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In mathematics, the '''spectrum of a [[C*-algebra]]''' or '''dual of a C*-algebra''' ''A'', denoted ''Â'', is the set of [[unitary equivalence]] classes of [[irreducible representation|irreducible]] *-representations of ''A''. A [[*-representation]] π of ''A'' on a [[Hilbert space]] ''H'' is '''irreducible''' if, and only if, there is no closed subspace ''K'' different from ''H'' and {0} which is invariant under all operators π(''x'') with ''x'' ∈ ''A''. We implicitly assume that irreducible representation means ''non-null'' irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-[[dimension]]al [[space (mathematics)|spaces]].  As explained below, the spectrum ''Â'' is also naturally a [[topological space]]; this generalizes the notion of the [[spectrum of a ring]].
{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
{{For|electromagnetic equations|Maxwell's equations}}
'''Maxwell's relations''' are a set of equations in [[thermodynamics]] which are derivable from the definitions of the [[thermodynamic potentials]]. These relations are named for the nineteenth-century physicist [[James Clerk Maxwell]].


One of the most important applications of this concept is to provide a notion of [[duality (mathematics)|dual]] object for any [[locally compact group]].  This dual object is suitable for formulating a [[Fourier transform]] and a [[Plancherel theorem]] for [[unimodular group|unimodular]] [[separable space|separable]] locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I.  The resulting duality theory for locally compact groups is however much weaker than the [[Tannaka–Krein duality]] theory for [[compact topological group]]s or [[Pontryagin duality]] for locally compact ''abelian'' groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the  dual of any finite dimensional full matrix algebra M<sub>''n''</sub>('''C''') consists of a single point.
==Equation==


== Primitive spectrum ==
The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation of an [[analytic function]] of two variables is irrelevant. If Φ is a thermodynamic potential and ''x<sub>i</sub>'' and ''x<sub>j</sub>'' are two different [[Thermodynamic potential#Natural variables|natural variables]] for that potential, then the Maxwell relation for that potential and those variables is:


The [[topology]] of ''Â'' can be defined in several equivalent ways.  We first define it in terms of the '''primitive spectrum''' .
{{Equation box 1
|title = '''Maxwell's relations''' ''(general)''
|indent =:
|equation = <math>\frac{\partial }{\partial x_j}\left(\frac{\partial \Phi}{\partial x_i}\right)=
\frac{\partial }{\partial x_i}\left(\frac{\partial \Phi}{\partial x_j}\right)
</math>
|border colour = #50C878
|background colour = #ECFCF4}}


The primitive spectrum of ''A'' is the set of [[primitive ideal]]s Prim(''A'') of ''A'', where  a primitive ideal is the kernel of an irreducible *-representation. The set of primitive ideals is a [[topological space]] with the '''hull-kernel topology''' (or '''Jacobson topology''').  This is defined as follows: If ''X'' is a set of primitive ideals, its '''hull-kernel closure''' is
where the [[partial derivatives]] are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are ''n''(''n'' − 1)/2 possible Maxwell relations where ''n'' is the number of natural variables for that potential.


:<math> \overline{X} = \{\rho \in \operatorname{Prim}(A): \rho \supseteq \bigcap_{\pi \in X} \pi\}. </math>
== The four most common Maxwell relations ==


Hull-kernel closure is easily shown to be an [[idempotent]] operation, that is
The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable ([[temperature]] ''T''; or [[entropy]] ''S'') and their ''mechanical'' natural variable ([[pressure]] ''P''; or [[volume]] ''V''):


:<math> \overline{\overline{X}} = \overline{X},</math>
{{Equation box 1
|title = '''Maxwell's relations''' ''(common)''
|indent =:
|equation =  


and it can be shown to satisfy the [[Kuratowski closure axioms]]. As a consequence, it can be shown that there is a unique topology &tau; on Prim(''A'') such that the closure of a set ''X'' with  respect to &tau; is identical to the hull-kernel closure of ''X''.
<math> \begin{align}
+\left(\frac{\partial T}{\partial V}\right)_S &=& -\left(\frac{\partial P}{\partial S}\right)_V &=& \frac{\partial^2 U }{\partial S \partial V}\\


Since unitarily equivalent representations have the same kernel, the map &pi; {{mapsto}} ker(&pi;) factors through a [[surjective]] map
+\left(\frac{\partial T}{\partial P}\right)_S &=& +\left(\frac{\partial V}{\partial S}\right)_P &=& \frac{\partial^2 H }{\partial S \partial P}\\
+\left(\frac{\partial S}{\partial V}\right)_T &=& +\left(\frac{\partial P}{\partial T}\right)_V &=& -\frac{\partial^2 A }{\partial T \partial V}\\


:<math> \operatorname{k}: \hat{A} \rightarrow \operatorname{Prim}(A). </math>
-\left(\frac{\partial S}{\partial P}\right)_T &=& +\left(\frac{\partial V}{\partial T}\right)_P &=& \frac{\partial^2 G }{\partial T \partial P}
\end{align}\,\!</math>


We use the map ''k'' to define the topology on ''Â'' as follows:
|border colour = #0073CF
|background colour=#F5FFFA}}


'''Definition'''. The open sets of ''Â'' are inverse images ''k''<sup>&minus;1</sup>(''U'') of open subsets ''U'' of Prim(''A''). This is indeed a topology.
where the potentials as functions of their natural thermal and mechanical variables are the [[internal energy]] ''U''(''S, V''), [[Enthalpy]] ''H''(''S, P''), [[Helmholtz free energy]] ''A''(''T, V'') and [[Gibbs free energy]] ''G''(''T, P''). The [[thermodynamic square]] can be used as a [[mnemonic]] to recall and derive these relations.


The hull-kernel topology is an analogue for non-commutative rings of the [[Zariski topology]] for commutative rings.
=== Derivation ===


The topology on ''Â'' induced from the hull-kernel topology has other characterizations in terms of [[state (functional analysis)|state]]s of ''A''.
Maxwell relations are based on simple partial differentiation rules, in particular the [[Total derivative|total]] [[differential of a function]] and the symmetry of evaluating second order partial derivatives.


== Examples ==
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation
|-
|Derivation of the Maxwell relations can be deduced from the differential forms of the [[thermodynamic potentials]]:


=== Commutative C*-algebras ===
:<math>\begin{align}
[[File:3-dim commut algebra, subalgebras, ideals.svg|thumb|left|224px|3-dimensional  commutative C*-algebra and its ideals. Each of 8 ideals corresponds to a closed subset of discrete 3-points space (or to an open complement). Primitive ideals correspond to closed [[singleton (mathematics)|singletons]]. See details at the image description page.]]
dU &=& TdS-PdV \\
The spectrum of a commutative C*-algebra ''A'' coincides with the [[Gelfand transformation|usual dual]] of ''A'' (not to be confused with the [[Banach space|dual]] ''A''' of the Banach space ''A'').  In particular, suppose ''X'' is a [[compact space|compact]] [[Hausdorff space]].  Then there is a [[natural transformation|natural]] [[homeomorphism]]
dH &=& TdS+VdP \\
dA &=& -SdT-PdV \\
dG &=& -SdT+VdP \\
\end{align}\,\!</math>


:<math> \operatorname{I}: X \cong \operatorname{Prim}( \operatorname{C}(X)).</math>
These equations resemble [[Total derivative|total differentials]] of the form


This mapping is defined by
:<math>dz = \left(\frac{\partial z}{\partial x}\right)_y\!dx +
\left(\frac{\partial z}{\partial y}\right)_x\!dy</math>


: <math>  \operatorname{I}(x) = \{f \in \operatorname{C}(X): f(x) = 0 \}.</math>
And indeed, it can be shown for any equation of the form


I(''x'') is a closed maximal ideal in C(''X'') so is in fact primitive. For details of the proof, see the Dixmier reference.  For a commutative C*-algebra,
:<math>dz = Mdx + Ndy \,</math>


:<math> \hat{A} \cong \operatorname{Prim}(A).</math>
that


=== The C*-algebra of bounded operators ===
:<math>M = \left(\frac{\partial z}{\partial x}\right)_y, \quad
N = \left(\frac{\partial z}{\partial y}\right)_x</math>


Let ''H'' be a separable [[Hilbert space]]. ''L''(''H'') has two norm-closed *-ideals: ''I''<sub>0</sub>&nbsp;=&nbsp;{0} and the ideal ''K''&nbsp;=&nbsp;''K''(''H'') of compact operators.  Thus as a set, Prim(''L''(''H'')) =&nbsp;{''I''<sub>0</sub>,&nbsp;''K''}. Now
Consider, as an example, the equation <math>dH=TdS+VdP\,</math>. We can now immediately see that


* {''K''} is a closed subset of Prim(''L''(''H'')).
:<math>T = \left(\frac{\partial H}{\partial S}\right)_P, \quad
      V = \left(\frac{\partial H}{\partial P}\right)_S</math>


* The closure of {''I''<sub>0</sub>} is Prim(''L''(''H'')).
Since we also know that for functions with continuous second derivatives, the mixed partial derivatives are identical ([[Symmetry of second derivatives]]), that is, that


Thus Prim(''L''(''H'')) is a non-Hausdorff space.
:<math>\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)_y =
\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)_x =
\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}</math>


The spectrum of ''L''(''H'') on the other hand is much larger.  There are many inequivalent irreducible representations with kernel ''K''(''H'') or with kernel&nbsp;{0}.
we therefore can see that


=== Finite dimensional C*-algebras ===
:<math> \frac{\partial}{\partial P}\left(\frac{\partial H}{\partial S}\right)_P =
\frac{\partial}{\partial S}\left(\frac{\partial H}{\partial P}\right)_S </math>


Suppose ''A'' is a finite dimensional C*-algebra.  It is known ''A'' is isomorphic to a finite direct sum of full matrix algebras:
and therefore that


:<math> A \cong \bigoplus_{e \in \operatorname{min}(A)} A e, </math>
:<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P</math>


where min(''A'') are the minimal central projections of ''A''.  The spectrum of ''A'' is canonically isomorphic to min(''A'') with the [[discrete topology]]. For finite dimensional C*-algebras, we also have the isomorphism
Each of the four Maxwell relationships given above follows similarly from one of the [[Gibbs equations]].
|}


:<math> \hat{A} \cong \operatorname{Prim}(A).</math>
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Extended derivation
|-
|Combined form first and second law of thermodynamics,
:<math>TdS = dU+PdV</math> (Eq.1)


== Other characterizations of the spectrum ==
U, S, and V are state functions.
Let,
:<math>U = U(x,y)</math>
:<math>S = S(x,y)</math>
:<math>V = V(x,y)</math>
:<math>dU = \left(\frac{\partial U}{\partial x}\right)_y\!dx +
\left(\frac{\partial U}{\partial y}\right)_x\!dy</math>


The hull-kernel topology is easy to describe abstractly, but in practice for C*-algebras associated to [[locally compact]] [[topological group]]s, other characterizations of the topology on the spectrum in terms of positive definite functions are desirable.
:<math>dS = \left(\frac{\partial S}{\partial x}\right)_y\!dx +
\left(\frac{\partial S}{\partial y}\right)_x\!dy</math>


In fact, the topology on ''Â'' is intimately connected with the concept of [[weak containment]] of representations as is shown by the following:
:<math>dV = \left(\frac{\partial V}{\partial x}\right)_y\!dx +
\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>


'''Theorem'''.  Let ''S'' be subset of ''Â''. Then the following are equivalent for an irreducile representation &pi;
Substitute them in Eq.1 and one gets,
:<math>T\left(\frac{\partial S}{\partial x}\right)_y\!dx +
  T\left(\frac{\partial S}{\partial y}\right)_x\!dy = \left(\frac{\partial U}{\partial x}\right)_y\!dx +
\left(\frac{\partial U}{\partial y}\right)_x\!dy + P\left(\frac{\partial V}{\partial x}\right)_y\!dx +
P\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>


# The equivalence class of &pi; in ''Â'' is in the closure of ''S''
And also written as,
# Every state associated to &pi;, that is one of the form
:<math>\left(\frac{\partial U}{\partial x}\right)_y\!dx +
\left(\frac{\partial U}{\partial y}\right)_x\!dy = T\left(\frac{\partial S}{\partial x}\right)_y\!dx +
T\left(\frac{\partial S}{\partial y}\right)_x\!dy - P\left(\frac{\partial V}{\partial x}\right)_y\!dx -
P\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>


::<math> f_\xi(x) = \langle \xi  \mid \pi(x) \xi \rangle </math>
comparing the coefficient of dx and dy, one gets
:<math>\left(\frac{\partial U}{\partial x}\right)_y = T\left(\frac{\partial S}{\partial x}\right)_y - P\left(\frac{\partial V}{\partial x}\right)_y</math>
:<math>\left(\frac{\partial U}{\partial y}\right)_x = T\left(\frac{\partial S}{\partial y}\right)_x - P\left(\frac{\partial V}{\partial y}\right)_x</math>


:with ||&xi;||=1, is the weak limit of states associated to representations in ''S''.
Differentiating above equations by y, x respectively<br />
:<math>\left(\frac{\partial^2U}{\partial y\partial x}\right) = \left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y + T\left(\frac{\partial^2 S}{\partial y\partial x}\right) - \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y - P\left(\frac{\partial^2 V}{\partial y\partial x}\right)</math> (Eq.2)
:and
:<math>\left(\frac{\partial^2U}{\partial x\partial y}\right) = \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x + T\left(\frac{\partial^2 S}{\partial x\partial y}\right) - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x - P\left(\frac{\partial^2 V}{\partial x\partial y}\right)</math> (Eq.3)


The second condition means exactly that &pi; is weakly contained in ''S''.
U, S, and V are exact differentials, therefore,
:<math>\left(\frac{\partial^2U}{\partial y\partial x}\right) = \left(\frac{\partial^2U}{\partial x\partial y}\right)</math>
:<math>\left(\frac{\partial^2S}{\partial y\partial x}\right) = \left(\frac{\partial^2S}{\partial x\partial y}\right)
:\left(\frac{\partial^2V}{\partial y\partial x}\right) = \left(\frac{\partial^2V}{\partial x\partial y}\right)</math>


The [[GNS construction]] is a recipe for associating states of a C*-algebra ''A'' to representations of ''A''.  By one of the basic theorems associated to the GNS construction, a state ''f'' is [[pure state|pure]] if and only if the associated representation &pi;<sub>''f''</sub> is irreducible. Moreover, the mapping &kappa;: PureState(''A'') &rarr; ''Â'' defined by ''f'' {{mapsto}} &pi;<sub>''f''</sub> is a surjective map.
Subtract eqn(2) and (3) and one gets<br />
:<math>\left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y - \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y = \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x</math>
:''Note: The above is called the general expression for Maxwell's thermodynamical relation.''


From the previous theorem one can easily prove the following;
;Maxwell's first relation
:Allow x = S and y = V and one gets
:<math>\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V</math>


'''Theorem''' The mapping
;Maxwell's second relation
:Allow x = T and y = V and one gets
:<math>\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V</math>


: <math> \kappa: \operatorname{PureState}(A) \rightarrow \hat{A} </math>
;Maxwell's third relation
:Allow x = S and y = P and one gets
:<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P</math>


given by the GNS construction is continuous and open.
;Maxwell's fourth relation
:Allow x = T and y = P and one gets
:<math>\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P</math>


=== The space Irr<sub>''n''</sub>(''A'') ===
;Maxwell's fifth relation
:Allow x = P and y = V
:<math>\left(\frac{\partial T}{\partial P}\right)_V \left(\frac{\partial S}{\partial V}\right)_P</math><math>-\left(\frac{\partial T}{\partial V}\right)_P \left(\frac{\partial S}{\partial P}\right)_V</math> = 1


There is yet another characterization of the topology on ''Â'' which arises by considering the space of representations as a topological space with an appropriate pointwise convergence topology.  More precisely, let ''n'' be a cardinal number and let ''H''<sub>''n''</sub> be the canonical Hilbert space of dimension ''n''.
;Maxwell's sixth relation
:Allow x = T and y = S and one gets
:<math>\left(\frac{\partial P}{\partial T}\right)_S \left(\frac{\partial V}{\partial S}\right)_T -\left(\frac{\partial P}{\partial S}\right)_T \left(\frac{\partial V}{\partial T}\right)_S</math> = 1
|}


Irr<sub>''n''</sub>(''A'') is the space of irreducible *-representations of ''A'' on ''H''<sub>''n''</sub> with the point-weak topology.  In terms of convergence of nets, this topology is defined by &pi;<sub>''i''</sub> &rarr; &pi; if and only if
== General Maxwell relationships ==


:<math> \langle \pi_i(x) \xi \mid \eta \rangle \rightarrow \langle \pi(x) \xi \mid \eta \rangle \quad \forall \xi, \eta \in H_n \ x \in A. </math>
The above are not the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the [[Particle number|number of particles]] is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles ''N''&nbsp; is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be:


It turns out that this topology on Irr<sub>''n''</sub>(''A'') is the same as the point-strong topology, i.e. &pi;<sub>''i''</sub> &rarr; &pi; if and only if
:<math>
\left(\frac{\partial \mu}{\partial P}\right)_{S, N} =
\left(\frac{\partial V}{\partial N}\right)_{S, P}\qquad=
\frac{\partial^2 H }{\partial P \partial N}
</math>


:<math> \pi_i(x) \xi  \rightarrow \pi(x) \xi  \quad \mbox{ normwise } \forall \xi \in H_n \ x \in A. </math>
where μ is the [[chemical potential]]. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations.


'''Theorem'''. Let ''Â''<sub>''n''</sub> be the subset of ''Â'' consisting of equivalence classes of representations whose underlying  Hilbert space has dimension ''n''.  The canonical map Irr<sub>''n''</sub>(''A'') &rarr; ''Â''<sub>''n''</sub> is continuous and open. In particular, ''Â''<sub>''n''</sub> can be regarded as the quotient topological space of Irr<sub>''n''</sub>(''A'') under unitary equivalence.
Each equation can be re-expressed using the relationship


'''Remark'''.  The piecing together of the various ''Â''<sub>''n''</sub> can be quite complicated.
:<math>\left(\frac{\partial y}{\partial x}\right)_z
=
1\left/\left(\frac{\partial x}{\partial y}\right)_z\right.</math>


== Mackey Borel structure ==
which are sometimes also known as Maxwell relations.


''Â'' is a topological space and thus can also be regarded as a [[Borel set|Borel space]].  A famous conjecture of [[G. Mackey]] proposed that a ''separable'' locally compact group is of type I if and only if the Borel space is standard, i.e. is isomorphic (in the category of Borel spaces) to the underlying Borel space of a [[Polish space|complete separable metric space]].  Mackey called Borel spaces with this property '''smooth'''. This conjecture was proved by [[James Glimm]] for separable C*-algebras in the 1961 paper listed in the references below.
== See also ==
* [[Table of thermodynamic equations]]
* [[Thermodynamic equations]]


'''Definition'''. A non-degenerate *-representation &pi; of a separable C*-algebra ''A'' is a '''factor representation''' if and only if the center of the von Neumann algebra generated by &pi;(''A'') is one-dimensional.  A C*-algebra ''A'' is of type I if and only if any separable factor representation of ''A'' is a finite or countable multiple of an irreducible one.
==External links==
*http://theory.ph.man.ac.uk/~judith/stat_therm/node48.html a partial derivation of Maxwell's relations


Examples of separable locally compact groups ''G'' such that C*(''G'') is of type I are [[connected space|connected]] (real) [[nilpotent]] [[Lie group]]s and connected real [[semi-simple]] Lie groups.  Thus the [[Heisenberg group]]s are all of type I. Compact and abelian groups are also of type I.


'''Theorem'''. If ''A'' is separable, ''Â'' is smooth if and only if ''A'' is of type I.
[[Category:Thermodynamics]]
[[Category:Concepts in physics]]
[[Category:James Clerk Maxwell]]
[[Category:Thermodynamic equations]]


The result implies a far-reaching generalization of the structure of representations of separable type I C*-algebras and correspondingly of separable locally compact groups of type I.
[[ar:علاقات ماكسويل]]
 
[[ca:Relacions de Maxwell]]
== Algebraic primitive spectra  ==
[[cs:Maxwellovy relace]]
 
[[de:Maxwell-Beziehung]]
Since a C*-algebra ''A'' is a [[ring (mathematics)|ring]], we can also consider the set of [[primitive ideal]]s of ''A'', where ''A'' is regarded algebraically.  For a ring an ideal is primitive if and only if it is the [[Annihilator (ring theory)|annihilator]] of a [[simple module]].  It turns out that for a C*-algebra ''A'', an ideal is algebraically primitive [[if and only if]] it is primitive in the sense defined above.
[[fa:روابط ماکسول (ترمودینامیک)]]
 
[[fr:Relations de Maxwell]]
'''Theorem'''.  Let ''A'' be a C*-algebra.  Any algebraically irreducible representation of ''A'' on a complex vector space is algebraically equivalent to a topologically irreducible *-representation on a Hilbert space.  Topologically irreducible *-representations on a Hilbert space are algebraically isomorphic if and only if they are unitarily equivalent.
[[ko:맥스웰 관계식]]
 
[[it:Relazioni di Maxwell]]
This is the Corollary of Theorem 2.9.5 of the Dixmier reference.
[[he:קשרי מקסוול]]
 
[[ja:マクスウェルの関係式]]
If ''G'' is a locally compact group, the topology on dual space of the [[group algebra|group C*-algebra]] C*(''G'') of ''G'' is called the '''Fell topology''', named after [[J. M. G. Fell]].
[[pl:Relacje Maxwella]]
 
[[sv:Maxwells relationer]]
== References ==
[[th:ความสัมพันธ์ของแมกซ์เวลล์]]
* J.  Dixmier, ''Les C*-algèbres et leurs représentations'', Gauthier-Villars, 1969.
[[zh:麦克斯韦关系式]]
* J. Glimm, ''Type I C*-algebras'', Annals of Mathematics, vol 73, 1961.
* G. Mackey, ''The Theory of Group Representations'', The University of Chicago Press, 1955.
 
{{Functional Analysis}}
 
{{DEFAULTSORT:Spectrum of a C-algebra}}
[[Category:C*-algebras]]
[[Category:Spectral theory]]

Revision as of 23:59, 12 August 2014

The writer is called Wilber Pegues. Distributing manufacturing is how he makes a living. Mississippi is the only place I've been residing in but I will have to move in a year or two. To climb is some thing she would never give up.

my web-site; cheap psychic readings [netwk.hannam.ac.kr] 28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. These relations are named for the nineteenth-century physicist James Clerk Maxwell.

Equation

The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant. If Φ is a thermodynamic potential and xi and xj are two different natural variables for that potential, then the Maxwell relation for that potential and those variables is:

The Aharonov–Casher effect is a quantum mechanical phenomenon in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It is used for example to observe interference of neutrons or fluxons.

In both effects the particle acquires some phase shift (φ) while traveling along some path P. In the Aharonov–Bohm effect it is

φAB=qℏ∫PA⋅dx

While for the Aharonov–Casher effect it is

φAC=1ℏc2∫P(E×μ)⋅dx

where q is its charge and μ is the magnetic moment.

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    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Template:Cite paper

See also


Template:Quantum-stub

where the partial derivatives are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are n(n − 1)/2 possible Maxwell relations where n is the number of natural variables for that potential.

The four most common Maxwell relations

The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature T; or entropy S) and their mechanical natural variable (pressure P; or volume V):

The Aharonov–Casher effect is a quantum mechanical phenomenon in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It is used for example to observe interference of neutrons or fluxons.

In both effects the particle acquires some phase shift (φ) while traveling along some path P. In the Aharonov–Bohm effect it is

φAB=qℏ∫PA⋅dx

While for the Aharonov–Casher effect it is

φAC=1ℏc2∫P(E×μ)⋅dx

where q is its charge and μ is the magnetic moment.

Bibliography

  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  • Template:Cite paper

See also


Template:Quantum-stub

where the potentials as functions of their natural thermal and mechanical variables are the internal energy U(S, V), Enthalpy H(S, P), Helmholtz free energy A(T, V) and Gibbs free energy G(T, P). The thermodynamic square can be used as a mnemonic to recall and derive these relations.

Derivation

Maxwell relations are based on simple partial differentiation rules, in particular the total differential of a function and the symmetry of evaluating second order partial derivatives.

General Maxwell relationships

The above are not the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the number of particles is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles N  is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be:

(∂μ∂P)S,N=(∂V∂N)S,P=∂2H∂P∂N

where μ is the chemical potential. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations.

Each equation can be re-expressed using the relationship

(∂y∂x)z=1/(∂x∂y)z

which are sometimes also known as Maxwell relations.

See also

External links

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