Canonical ensemble

The canonical ensemble is a statistical ensemble that corresponds to a physical system that exchanges energy with the environment (thermostat), being in thermal equilibrium with it, but does not exchange matter , since it is separated from the thermostat by a particle-tight partition. The parameters of the abbreviated description of such a system are the number of particles $N$ ${\ displaystyle N}$ $N$ and average energy ${\bar {E}}$ ${\ displaystyle {\ bar {E}}}$ ${\ displaystyle {\ bar {E}}}$ .

Gibbs Distribution

The canonical ensemble includes microscopic states with different energies. The probability of this particular state with energy $E_{\tau }$ ${\ displaystyle E _ {\ tau}}$ $E_{\tau }$ depends only on the energy value and is given by the Gibbs distribution

w_{\tau }={\frac {1}{Z}}e^{-E_{\tau }/k_{B}T}

{\ displaystyle w _ {\ tau} = {\ frac {1} {Z}} e ^ {- E _ {\ tau} / k_ {B} T}}

w_{\tau }={\frac {1}{Z}}e^{-E_{\tau }/k_{B}T}

,

where Z is the normalization constant, which is selected from the condition that the sum of the probabilities is 1.

Z=\sum _{\tau }e^{-E_{\tau }/k_{B}T}

{\ displaystyle Z = \ sum _ {\ tau} e ^ {- E _ {\ tau} / k_ {B} T}}

Z=\sum _{\tau }e^{-E_{\tau }/k_{B}T}

.

Z is called the partition function .

Classic Case

The volume of phase space occupied by the canonical ensemble of $N$ ${\ displaystyle N}$ $N$ identical particles, called the partition function , which is given by the formula.

Z_{N}={\frac {1}{N!}}\int {\frac {d^{3N}pd^{3N}q}{h^{3N}}}\exp[-\beta H(p,q)]

{\ displaystyle Z_ {N} = {\ frac {1} {N!}} \ int {\ frac {d ^ {3N} pd ^ {3N} q} {h ^ {3N}}} \ exp [- \ beta H (p, q)]}

Z_{N}={\frac {1}{N!}}\int {\frac {d^{3N}pd^{3N}q}{h^{3N}}}\exp[-\beta H(p,q)]

Where $\beta =1/k_{B}T$ ${\ displaystyle \ beta = 1 / k_ {B} T}$ $\beta =1/k_{B}T$ . Compliance with the general case: $\tau \to (p,q)$ ${\ displaystyle \ tau \ to (p, q)}$ $\tau \to (p,q)$ , $\sum _{\tau }\to \int {\frac {d^{3N}pd^{3N}q}{h^{3N}}}$ ${\ displaystyle \ sum _ {\ tau} \ to \ int {\ frac {d ^ {3N} pd ^ {3N} q} {h ^ {3N}}}}$ $\sum _{\tau }\to \int {\frac {d^{3N}pd^{3N}q}{h^{3N}}}$ but $E_{\tau }\to H(p,q)$ ${\ displaystyle E _ {\ tau} \ to H (p, q)}$ $E_{\tau }\to H(p,q)$ . Factor $1/N!$ ${\ displaystyle 1 / N!}$ $1/N!$ appears in accordance with the principle of indistinguishability of particles .

Literature

Hill T. Statistical Mechanics, Principles, and Selected Applications. - M.: IL, 1960.

Canonical ensemble

Gibbs Distribution

Classic Case

Literature

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