Ensemble average (statistical mechanics)

Short description: Mean of a quantity that is a function of the microstate of a system

In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the microstate of a system, according to the distribution of the system on its micro-states in this ensemble.

Since the ensemble average is dependent on the ensemble chosen, its mathematical expression varies from ensemble to ensemble. However, the mean obtained for a given physical quantity doesn't depend on the ensemble chosen at the thermodynamic limit. The grand canonical ensemble is an example of an open system.

## Contents

* 1 Canonical ensemble average
* 1.1 Classical statistical mechanics
* 1.2 Quantum statistical mechanics
* 2 Ensemble average in other ensembles
* 2.1 Microcanonical ensemble
* 2.2 Canonical ensemble
* 2.3 Grand canonical ensemble
* 3 See also

## Canonical ensemble average

### Classical statistical mechanics

For a classical system in thermal equilibrium with its environment, the ensemble average takes the form of an integral over the phase space of the system:

[math]\displaystyle{ \bar{A}=\frac{\int{Ae^{-\beta H(q_1, q_2, ... q_M, p_1, p_2, ... p_N)}d\tau}}{\int{e^{-\beta H(q_1, q_2, ... q_M, p_1, p_2, ... p_N)}d\tau}} }[/math]

where:

[math]\displaystyle{ \bar{A} }[/math] is the ensemble average of the system property A,

[math]\displaystyle{ \beta }[/math] is [math]\displaystyle{ \frac {1}{kT} }[/math], known as thermodynamic beta,

H is the Hamiltonian of the classical system in terms of the set of coordinates [math]\displaystyle{ q_i }[/math] and their conjugate generalized momenta [math]\displaystyle{ p_i }[/math], and

[math]\displaystyle{ d\tau }[/math] is the volume element of the classical phase space of interest.

The denominator in this expression is known as the partition function, and is denoted by the letter Z.

### Quantum statistical mechanics

In quantum statistical mechanics, for a quantum system in thermal equilibrium with its environment, the weighted average takes the form of a sum over quantum energy states, rather than a continuous integral:

[math]\displaystyle{ \bar{A}=\frac{\sum_i{A_ie^{-\beta E_i}}}{\sum_i{e^{-\beta E_i}}} }[/math]

## Ensemble average in other ensembles

The generalized version of the partition function provides the complete framework for working with ensemble averages in thermodynamics, information theory, statistical mechanics and quantum mechanics.

### Microcanonical ensemble

The microcanonical ensemble represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all constant.

### Canonical ensemble

The canonical ensemble represents a closed system which can exchange energy (E) with its surroundings (usually a heat bath), but the volume (V) and the number of particles (N) are all constant.

### Grand canonical ensemble

The grand canonical ensemble represents an open system which can exchange energy (E) as well as particles with its surroundings but the volume (V) is kept constant.

## See also

* Maxwell–Boltzmann statistics

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Statistical mechanics

Theory|

* Principle of maximum entropy
* ergodic theory

Statistical thermodynamics|

* Ensembles
* partition functions
* equations of state
* thermodynamic potential:
* U
* H
* F
* G
* Maxwell relations

Models|

* Ferromagnetism models
* Ising
* Potts
* Heisenberg
* percolation
* Particles with force field
* depletion force
* Lennard-Jones potential

Mathematical approaches|

* Boltzmann equation
* H-theorem
* Vlasov equation
* BBGKY hierarchy
* stochastic process
* mean field theory and conformal field theory

Critical phenomena|

* Phase transition
* Critical exponents
* correlation length
* size scaling

Entropy|

* Boltzmann
* Shannon
* Tsallis
* Rényi
* von Neumann

Applications|

* Statistical field theory
* elementary particle
* superfluidity
* condensed matter physics
* complex system
* chaos
* information theory
* Boltzmann machine

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