Ensemble average
Encyclopedia
In statistical mechanics
, the ensemble average is defined as the mean
of a quantity that is a function of the micro-state of a system (the ensemble
of possible states), 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
.
with its environment, the ensemble average takes the form of an integral over the phase space
of the system:
is the ensemble average of the system property A,
is 
, known as thermodynamic beta,
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.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, the ensemble average is defined as the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of a quantity that is a function of the micro-state of a system (the ensemble
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
of possible states), according to the distribution of the system on its micro-states in this ensemble
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
.
Since the ensemble average is dependent on the ensemble
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
chosen, its mathematical expression varies from ensemble to ensemble. However, the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
obtained for a given physical quantity doesn't depend on the ensemble chosen at the thermodynamic limit
Thermodynamic limit
In thermodynamics, particularly statistical mechanics, the thermodynamic limit is reached as the number of particles in a system, N, approaches infinity...
.
Classical statistical mechanics
For a classical system in thermal equilibriumThermal equilibrium
Thermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
with its environment, the ensemble average takes the form of an integral over the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of the system:
- where:
is the ensemble average of the system property A, is , known as thermodynamic beta,- H is the Hamiltonian of the classical system in terms of the set of coordinates
and their conjugate generalized momenta 
, and
is the volume elementVolume element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates...
of the classical phase space of interest.
The denominator in this expression is known as the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
, and is denoted by the letter Z.