Introduction to Mathematical Physics/N body problems and statistical equilibrium/Quantum perfect gases

Introduction

Consider a quantum perfect gas, {\it i. e.}, a system of independent particle that have to be described using quantum mechanics, \index{quantum perfect gas} in equilibrium with a particle reservoir [ph:physt:Diu89] and with a thermostat. The description adopted is called "grand--canonical". It can be shown that the solution of the entropy maximalization problem (using Lagrange multipliers method) provides the occupation probability $P_{l}$ of a state $l$ characterized by an energy $E_{l}$ an a number of particle $N_{l}$:

$P_{l}={\frac {1}{Z}}e^{-\beta E_{l}+\mu \beta N_{l}}$

with

$Z=\sum _{l}e^{-\beta E_{l}+\mu \beta N_{l}}$

Assume that the independent particles that constitute the system are in addition identical undiscernible. State $l$ can then be defined by the datum of the states $\lambda$ of individual particles.

figoccup

A system whose number of independent particles $N_{l}$ varies can be described by the set of $N_{l}$ particles energies or, in an equivalent way, by numbers $N_{\lambda }$ of particles at the various energy levels $\epsilon _{\lambda }$.

Let $\epsilon _{\lambda }$ be the energy of a particle in the state $\lambda$. The energy of the system in state $l$ is then:

$E_{l}=\sum _{\lambda }N_{\lambda }\epsilon _{\lambda }$

where

$N_{\lambda }$

is the number of particles that are in a state of energy

$E_{\lambda }$.

This number is called occupation number of state of energy\index{occupation number} $E_{\lambda }$ (see figure figoccup). Thus:

$N_{l}=\sum _{\lambda }N_{\lambda }$

Partition function becomes:

$Z=\Pi _{\lambda }\xi _{\lambda }$

with

$\xi _{\lambda }=\sum _{N_{l}}e^{-\beta N_{\lambda }\epsilon _{\lambda }+\beta N_{\lambda }\mu }$

The average particles number in the system is given by:

$\beta {\bar {N}}=-{\frac {\partial \ln Z}{\partial \mu }}$

that can be written:

$\beta {\bar {N}}=\sum _{\lambda }N_{\lambda }$

where $N_{\lambda }$ represents the average occupation number and is defined by following equality:

$\beta {\bar {N}}_{\lambda }={\frac {\partial \ln \xi _{\lambda }}{\partial \mu }}$

Fermion gases

If particles are fermions\index{fermions}, from Pauli principle\index{Pauli principle}, occupation number $N_{\lambda }$ value can only be either zero (there is no particle in state with energy $\epsilon _{\lambda }$) or one (a unique particle has an energy $\epsilon _{\lambda }$). The expression of partition function then allows to evaluate the various thermodynamical properties of the considered system. An application example of this formalism is the study of electrical properties of semiconductors [ph:physt:Diu89] \index{metal}\index{semi-conductor}. Fermion gases can also be used to model white dwarfs \index{white dwarf}. A white dwarf is a star [ph:physt:Diu89] very dense: its mass is of the order of an ordinary star's mass \index{star} (as sun), but its radius is 50 to 100 times smaller. Gravitational pressure implies star contraction. This pressure in an ordinary star is compensated by thermonuclear reaction that occur in the centre of the star. But is a white wharf, such reactions do not occur no more. Moreover, one can show that speed of electrons of the star is very small. As all electrons can not be is the same state from Pauli principle , they thus exert a pressure. This pressure called "quantum pressure" compensates gravitational pressure and avoid the star to collapse completely.

Boson gases

If particles are bosons, from Pauli principle, occupation number $N_{\lambda }$ can have any positive or zero value:

$\xi _{\lambda }^{B}=\sum _{N_{\lambda }=0}^{+\infty }e^{-\beta N_{\lambda }\epsilon _{\lambda }+\beta N_{\lambda }\mu }$

$\xi _{\lambda }^{B}$ is sum of geometrical series of reason:

$e^{-\beta \epsilon _{\lambda }+\beta \mu }$

where $\epsilon _{\lambda }$ is fixed. Series converges if

$\epsilon _{\lambda }-\mu >0$

It can be shown [ph:physt:Diu89] that at low temperatures, bosons gather in the state of lowest energy. This phenomena is called Bose condensation.\index{Bose condensation}

Remark:

Photons are bosons whose number is not conserved. This confers them a very peculiar behaviour: their chemical potential is zero.