Ensemble average

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Ensemble average

In statistical mechanics

Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
, 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

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 physics and physical chemistry, the thermodynamic limit is reached as the number of particles in a system N approaches infinity ? or in practical terms, one mole or Avogadro's number ? 6 x 1023....
.

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Encyclopedia

In statistical mechanics

Statistical mechanics

Mean

Statistical mechanics

Mean

Thermodynamic limit

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

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

Thermodynamic beta

In statistical mechanics, the thermodynamic beta is a numerical quantity related to the thermodynamic temperature of a system. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and thermodynamics....
,

H is the Hamiltonian of the classical system in terms of the set of coordinates and their conjugate generalized momenta , and

is the volume element of the classical phase space of interest.

The denominator in this expression is known as the partition function

Partition function

Partition function may refer to:*Partition function *Partition function , which generalizes its use in statistical mechanics and quantum field theory:...
, and is denoted by the letter Z.

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:

Ensemble average

Canonical ensemble average

Classical statistical mechanics

Quantum statistical mechanics

Characterization of the classical limit