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Prove that, for any system in equilibrium with a reservoir at temperature T, the average value of the energy is

E¯=-1ZZβ=-βlnZ

where β=1/kT. These formulas can be extremely useful when you have an explicit formula for the partition function.

Short Answer

Expert verified

Hence proved,

-1ZZβ=E¯

Step by step solution

01

Given information

For any system in equilibrium with a reservoir at temperature T, the average value of the energy is

E¯=-1ZZβ=-βlnZ

02

Explanation

The partition function is given by:

Z=se-βE(s)

Where βis Boltzmann's constant and is given by β=1kT

By partial derivative of partition equation with respect to β

Zβ=s-E(s)e-βE(s)

Multiplying by a factor of -1/Z

-1ZZβ=sE(s)e-βE(s)Z

The probability is given as:

P=1Ze-βE(s)

Equation (1) becomes,

role="math" localid="1647371210348" -1ZZβ=sE(s)P(s)

The average energy is:

role="math" localid="1647371238971" E¯=sE(s)P(s)

Therefore,

-1ZZβ=E¯

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