Consider N N N m m m V V V thermodynamic system
). A classical microstate
is ( q 1 , … , q N , p 1 , … , p N ) (q_1,\dots,q_N,p_1,\dots,p_N) ( q 1 , … , q N , p 1 , … , p N )
H ( p , q ) = ∑ i = 1 N ∣ p i ∣ 2 2 m . H(p,q)=\sum_{i=1}^N \frac{|p_i|^2}{2m}. H ( p , q ) = i = 1 ∑ N 2 m ∣ p i ∣ 2 . In the canonical ensemble
at inverse temperature β = 1 / ( k B T ) \beta=1/(k_B T) β = 1/ ( k B T ) N N N partition function
is
Z N ( β ) = 1 N ! h 3 N ∫ V N d q ∫ R 3 N d p e − β H ( p , q ) = 1 N ! ( V λ T 3 ) N ,
Z_N(\beta)=\frac{1}{N!\,h^{3N}}\int_{V^N}\!dq\int_{\mathbb R^{3N}}\!dp\;e^{-\beta H(p,q)}
= \frac{1}{N!}\left(\frac{V}{\lambda_T^3}\right)^N,
Z N ( β ) = N ! h 3 N 1 ∫ V N d q ∫ R 3 N d p e − β H ( p , q ) = N ! 1 ( λ T 3 V ) N , where the thermal wavelength is
λ T = h 2 π m k B T . \lambda_T=\frac{h}{\sqrt{2\pi m k_B T}}. λ T = 2 πm k B T h . Key thermodynamic results:
Helmholtz free energy. Using Helmholtz free energy
,
F ( T , V , N ) = − k B T log Z N . F(T,V,N)=-k_B T\log Z_N. F ( T , V , N ) = − k B T log Z N . Equation of state (ideal gas law).
p = − ( ∂ F ∂ V ) T , N = N k B T V , so p V = N k B T . p = -\left(\frac{\partial F}{\partial V}\right)_{T,N} = \frac{N k_B T}{V}, \qquad\text{so } pV=Nk_BT. p = − ( ∂ V ∂ F ) T , N = V N k B T , so p V = N k B T . Here p p p pressure
.
Internal energy and heat capacity.
U = − ∂ ∂ β log Z N = 3 2 N k B T , U = -\frac{\partial}{\partial\beta}\log Z_N = \frac{3}{2}N k_B T, U = − ∂ β ∂ log Z N = 2 3 N k B T , hence
C V = ( ∂ U ∂ T ) V , N = 3 2 N k B , C_V=\left(\frac{\partial U}{\partial T}\right)_{V,N}=\frac{3}{2}N k_B, C V = ( ∂ T ∂ U ) V , N = 2 3 N k B , matching heat capacity at constant volume
.
Entropy. From S = − ( ∂ F / ∂ T ) V , N S=-(\partial F/\partial T)_{V,N} S = − ( ∂ F / ∂ T ) V , N entropy
),
S = N k B [ ln ( V N λ T 3 ) + 5 2 ] , S = N k_B\left[\ln\!\left(\frac{V}{N\lambda_T^3}\right)+\frac{5}{2}\right], S = N k B [ ln ( N λ T 3 V ) + 2 5 ] , which is the Sackur–Tetrode formula
in its T T T
Energy fluctuations (canonical). The canonical variance satisfies
V a r ( U ) = k B T 2 C V , \mathrm{Var}(U)=k_B T^2 C_V, Var ( U ) = k B T 2 C V , relating variance
to thermodynamic response.
Prerequisites: canonical ensemble
, canonical partition function
, Helmholtz free energy
, thermodynamic entropy
.