Ensemble average Totally Explained

Everything about Ensemble Average totally explained

In statistical mechanics, the ensemble average is defined as the mean of a quantity that's 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.

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:
ť

ar

, known as thermodynamic beta,

ť H is the Hamiltonian (or energy function) of the classical system in terms of the set of coordinates

q_i

and their conjugate generalized momenta

p_i

, and

ť

d	au

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

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:

Characterization of the classical limit

Ensemble average in other ensembles

Microcanonical ensemble

Macrocanonical ensemble

Further Information

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