### Statistical Mechanics Lecture 22 of 29

July 11, 2012 by Matteo Marsili

M. Marsili, ICTP

This lesson began with a review of how the partition function (Z) for the classical ideal gas was obtained. Onece the partition function is obtained it can be calculated the Helmholtz free energy F = -KT log[Z] . A key point is the N! factor that takes into account the indistinguishability of particles. A question led to a discussion of a gas made of two distinguishable components. It was shown how a straightforward generalization of the method used for the single component ideal gas leads to the correct result also in the multicomponent ideal gas.

It was shown in the previous lecture that the microcanonical ensemble and canonical ensemble lead to the same ideal gas thermodynamics. The explanation of this fact is related to a general feature of statistical systems, namely, the fact that fluctuations become irrelevant as the system grows in size.

The energy is not fixed in the canonical ensemble, only the average energy is. One can calculate very generally the size of the energy fluctuations as the variance of the energy. The standard result was obtained <? E2> = kT2CV. where CV is the specific heat. From here we find that the relative
fluctuation, (<? E2>)1/2/E scales with the number of particles as N-1/2. Thus, if N is very large, the relative fluctuation is negligible and therefore, a system in which the energy is not fixed (canonical)
behaves in all respects as if its energy was fixed to its average value (microcanonical). This explains why both ensembles lead to the same thermodynamics.

In the secound part of the lesson the Equipartition theorem for classical systems was introduced a general formula that relates the temperature of a system with its average energies. The equipartition theorem shows that in thermal equilibrium, any degree of freedom which appears only quadratically in the energy has an average energy of <span class="frac nowrap">1?2</span>kBT and therefore contributes <span class="frac nowrap">1?2</span>kB to the system's specific heat.

Lecture 7 | Modern Physics: Statistical Mechanics

May 11, 2009 - Leonard Susskind lectures on harmonic oscillators, quantum states, boxes of radiation and all associated computations such as wavelengths, volume, energy and temperature.

As a complementary tool you can also see some lessons on Statistical Mechanics given in the <span class="yt-user-name author">Stanford</span> University.

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