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| In [[statistical mechanics]], a '''microstate''' is a specific microscopic configuration of a [[thermodynamic system]] that the system may occupy with a certain probability in the course of its [[thermal fluctuations]]. In contrast, the '''macrostate''' of a system refers to its macroscopic properties, such as its [[temperature]] and [[pressure]].<ref>[http://khanexercises.appspot.com/video?v=5EU-y1VF7g4 Macrostates and Microstates]</ref> Treatments on [[statistical mechanics]], <ref>{{cite book|title=Fundamentals of Statistical and Thermal Physics| last=Reif| first=Frederick| year=1965| publisher=McGraw-Hill| isbn=07-051800-9| page=66|url=http://books.google.co.in/books/about/Fundamentals_of_statistical_and_thermal.html?id=w5dRAAAAMAAJ&redir_esc=y}}</ref> <ref>{{cite book|title=Statistical Mechanics| last=Pathria| first=R K| year=1965| publisher=Butterworth-Heinemann| isbn=0-7506-2469-8| page=10|url=http://books.google.co.in/books/about/Statistical_Mechanics.html?id=PIk9sF9j2oUC}}</ref> often give an equivalent definition of a macrostate by specifying it for an isolated thermodynamic system. A particular set of values of energy, number of particles and volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate.
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| A macrostate is characterized by a [[probability distribution]] of possible states across a certain [[Statistical ensemble (mathematical physics)|statistical ensemble]] of all microstates. This distribution describes the [[probability]] of finding the system in a certain microstate. In the [[thermodynamic limit]], the microstates visited by a macroscopic system during its fluctuations all have the same macroscopic properties.
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| == Microscopic definitions of thermodynamic concepts==
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| Statistical mechanics links the empirical thermodynamic properties of a system to the statistical distribution of an ensemble of microstates. All macroscopic thermodynamic properties of a system may be calculated from the [[Partition_function_(statistical_mechanics)|partition function]] that sums the energy of all its microstates.
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| At any moment a system is distributed across an ensemble of ''N'' microstates, each denoted by ''i'', and having a probability of occupation ''p''<sub>i</sub>, and an energy <math>E_i</math>. These microstates form a discrete set as defined by [[quantum statistical mechanics]], and <math>E_i</math> is an [[energy level]] of the system.
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| ===Internal energy===
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| The internal energy of the macrostate is the [[mean]] over all microstates of the system's energy
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| :<math>U = \langle E \rangle = \sum_{i=1}^N p_i \,E_i\ .</math>
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| This is a microscopic statement of the notion of energy associated with the [[first law of thermodynamics]].
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| ===Entropy===
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| The absolute [[entropy]] exclusively depends on the probabilities of the microstates and is defined as
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| : <math>S = -k_B\,\sum_i p_i \ln \,p_i,</math>
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| where <math>k_B</math> is [[Boltzmann's constant]], for the [[canonical ensemble]]. For the [[microcanonical ensemble]], consisting of only those microstates with energy equal to the energy of the macrostate, this simplifies to
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| : <math>S = k_B\,\ln W</math>, | |
| where <math>W</math> is the number of microstates. This form for entropy appears on [[Ludwig Boltzmann]]'s gravestone in Vienna.
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| The [[second law of thermodynamics]] describes how the entropy of an isolated system changes in time. The [[third law of thermodynamics]] is consistent with this definition, since zero entropy means that the macrostate of the system reduces to a single microstate.
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| ===Heat and work===
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| [[Heat]] is the energy transfer associated with a disordered, microscopic action on the system, associated with jumps in [[energy level]]s of the system.
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| [[Work (thermodynamics)|Work]] is the energy transfer associated with an ordered, macroscopic action on the system. If this action acts very slowly then the [[Adiabatic theorem]] implies that this will not cause a jump in the energy level of the system. The internal energy of the system can only change due to a change of the energies of the system's energy levels.
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| The microscopic definitions of heat and work are the following:
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| :<math>\delta W = \sum_{i=1}^N p_i\,dE_i</math>
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| :<math>\delta Q = \sum_{i=1}^N E_i\,dp_i</math>
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| so that
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| :<math>~dU = \delta W + \delta Q.</math>
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| The two above definitions of heat and work are among the few expressions of [[statistical mechanics]] where the sum corresponding to the quantum case cannot be converted into an [[integral]] in the classical limit of a [[microstate continuum]]. The reason is that classical microstates are usually not defined in relation to a precise associated quantum microstate, which means that when work changes the energy associated to the energy levels of the system, the energy of classical microstates doesn't follow this change.
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| ==See also==
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| * [[Quantum statistical mechanics]]
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| * [[Degrees of freedom (physics and chemistry)]]
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| * [[Ergodic hypothesis]]
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| * [[Phase space]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://theory.ph.man.ac.uk/~judith/stat_therm/node57.html Some illustrations of microstates vs. macrostates]
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| [[Category:Statistical mechanics]]
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