State sum

The partition function is an essential tool in statistical physics . Because of the English term function partition the partition function will partition function called, although not with the partition function is to be confused from the combinatorics. ${\displaystyle Z}$

All thermodynamic quantities can be derived from a sum of states (the function, not the value) . If the particle numbers are large enough, the system can also be viewed as continuous and the sums of states can be formulated as integral states . ${\displaystyle N}$

Micro-canonical partition function

The micro-canonical partition function is used to describe a closed system with constant internal energy , volume and number of particles without exchange with the environment in thermodynamic equilibrium . The associated ensemble is called the micro-canonical ensemble . It should already be pointed out at this point that there are two different definitions for the micro-canonical sum of states: With one definition, all states with energy are summed up and with the other definition, only the states in the energy shell are summed up. ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle N}$ ${\displaystyle U}$ ${\displaystyle U}$

Countable states

First, those systems are considered that can be in one of a finite or countable number of micro-states (systems with uncountable / continuous states are discussed further below).

For such systems (in the first definition) the micro- canonical sum of states is given by the number of those micro-states of a closed system with a given energy , particle number and volume (and possibly other parameters) whose total energy is less than or equal : ${\displaystyle Z_{\mathrm {m} }(U,N,V)}$ ${\displaystyle \psi }$ ${\displaystyle U}$ ${\displaystyle N}$ ${\displaystyle V}$ ${\displaystyle E_{\psi }(N,V)}$ ${\displaystyle U}$

${\displaystyle Z_{\mathrm {m} }(U,N,V)=\!\!\!\sum _{E_{\psi }(N,V)\leq U}\!\!\!1.}$

In the second widespread definition of the micro-canonical partition function , it is given by the number of states whose energy lies in the interval : ${\displaystyle z_{\mathrm {m} }}$ ${\displaystyle E_{\psi }}$ ${\displaystyle [U,U+\Delta U]}$

${\displaystyle z_{\mathrm {m} }(U,N,V)=\!\!\!\sum _{U<E_{\psi }(N,V)<U+\Delta U}\!\!\!1=Z_{\mathrm {m} }(U+\Delta U,N,V)-Z_{\mathrm {m} }(U,N,V).}$

If the system is in equilibrium (i.e. in the state of maximum entropy), the probability of encountering a certain micro-state is : ${\displaystyle \psi }$

${\displaystyle P(\psi |U,N,V)={\begin{cases}{\frac {1}{z_{\mathrm {m} }(U,N,V)}}&{\mbox{falls }}E_{\psi }(N,V)=U,\\0&{\mbox{sonst.}}\\\end{cases}}}$

Continuous states

In classical mechanics systems are often considered, the microstate of which can change continuously. An example is the ideal gas . The space (also called phase space ) of an ideal gas consisting of particles has dimensions: dimensions for the spatial coordinates and for the momentum coordinates . Every point in the phase space corresponds to a state of the system with energy , where the Hamilton function of the system is with particle number and volume . Since the closed systems considered in microcanonics have constant energy, the permitted states in space result in a hypersurface on which the system can move. The sum of states for such a gas is the volume enclosed by this -hyper surface, which can be written as a state integral: ${\displaystyle \Gamma }$ ${\displaystyle N}$ ${\displaystyle 6N}$ ${\displaystyle 3N}$ ${\displaystyle 3N}$ ${\displaystyle (p,q)}$ ${\displaystyle \psi }$ ${\displaystyle E_{\psi }=H(p,q,N,V)}$ ${\displaystyle H(p,q,N,V)}$ ${\displaystyle N}$ ${\displaystyle V}$ ${\displaystyle \Gamma }$ ${\displaystyle H(p,q,N,V)=U}$

${\displaystyle Z_{\mathrm {m} }(U,N,V)\;=\int \limits _{H(p,q,N,V)\leq U}{\frac {\mathrm {d} ^{3N}p\;\mathrm {d} ^{3N}q}{h^{3N}N!}}=\int \limits _{\mathbb {R} ^{6N}}\theta (U-H(p,q,N,V)){\frac {\mathrm {d} ^{3N}p\;\mathrm {d} ^{3N}q}{h^{3N}N!}},}$

where is the Heaviside function . The density of states is thus determined by: ${\displaystyle \theta }$

${\displaystyle D:={\frac {\mathrm {d} Z_{\mathrm {m} }(U,N,V)}{\mathrm {d} U}}=\int \limits _{\mathbb {R} ^{6N}}\delta (U-H(p,q,N,V)){\frac {\mathrm {d} ^{3N}p\;\mathrm {d} ^{3N}q}{h^{3N}N!}}.}$

Here is the Dirac δ function . The following applies: ${\displaystyle \delta }$

${\displaystyle D={\frac {\mathrm {d} Z_{m}(U,N,V)}{\mathrm {d} U}}=\lim _{\Delta U\to 0}{\frac {z_{m}}{\Delta U}}.}$

The probability of finding the gas around a certain condition is: ${\displaystyle (p,q)}$

${\displaystyle \mathrm {d} P(p,q|U,N,V)={\frac {1}{z_{\mathrm {m} }(U,N,V)}}\delta (U-H(p,q,N,V)){\frac {\mathrm {d} ^{3N}p\;\mathrm {d} ^{3N}q}{h^{3N}N!}}}$

Another definition of the micro-canonical partition function is often found . It is then added or integrated via the energy shell from to around the - hypersurface of the system in the space. The shell has the width . The discrete variant is (as described above): ${\displaystyle U-\Delta U}$ ${\displaystyle U}$ ${\displaystyle U={\mbox{const}}}$ ${\displaystyle \Gamma }$ ${\displaystyle \Delta U}$

${\displaystyle z_{\mathrm {m} }(U,N,V)=\sum _{U-\Delta U\leq E_{\psi }(N,V)\leq U}1,}$

For continuous systems the partition function is then:

${\displaystyle z_{\mathrm {m} }(U,N,V)=\int \limits _{U-\Delta U\leq H(p,q,N,V)\leq U}{\frac {\mathrm {d} ^{3N}p\;\mathrm {d} ^{3N}q}{h^{3N}N!}}.}$

For the values ​​of and approach each other, since almost all states are in the outer shell. ${\displaystyle N\gg 1}$ ${\displaystyle Z_{\mathrm {m} }}$ ${\displaystyle z_{\mathrm {m} }}$

Canonical partition function

In the canonical ensemble , it is not the energy of the system that is given, but the temperature . This ensemble is also called the Gibbs ensemble (see also Canonical State ). The partition function is

${\displaystyle Z_{k}(N,V,T)=\sum _{i}\mathrm {e} ^{-{\frac {E_{i}}{k_{\mathrm {B} }T}}}}$

with the Boltzmann constant . The canonical partition function can be written equivalently as: ${\displaystyle k_{\mathrm {B} }}$

${\displaystyle Z_{k}(N,V,T)=\int \mathrm {d} E\,\sum _{i}\delta (E-E_{i})e^{-{\frac {E}{k_{\mathrm {B} }T}}}=\int \mathrm {d} E\,\rho (E)e^{-{\frac {E}{k_{\mathrm {B} }T}}}}$,

where is the density of states

${\displaystyle \rho (E):=\sum _{i}\delta (E-E_{i})}$

was introduced.

The occupation probability of a microstate is ${\displaystyle i}$

${\displaystyle p_{i}={\frac {1}{Z_{k}(N,V,T)}}\mathrm {e} ^{-{\frac {E_{i}}{k_{\mathrm {B} }T}}}.}$

The canonical state integral in three-dimensional space is

${\displaystyle Z_{k}(N,V,T)=\int \mathrm {e} ^{-{\frac {H(\mathbf {p,q} )}{k_{\mathrm {B} }T}}}\,{\frac {\mathrm {d} ^{N}\mathbf {p} \;\mathrm {d} ^{N}\mathbf {q} }{h^{3N}N!}}.}$

Where is the Hamilton function . The Gibbs factor comes from the indistinguishability of the particles. If one omitted this factor, one would instead have distinguishable states and compared to many microstates, which would result in Gibbs' paradox : two quantities of the same ideal gas separated by a partition have the same temperature and the same pressure . When pulling out the partition, an increase in entropy is wrongly observed without the factor. ${\displaystyle H}$ ${\displaystyle 1/N!}$ ${\displaystyle N}$ ${\displaystyle N!}$ ${\displaystyle 1/N!}$

Grand canonical partition function

In the grand canonical ensemble , the chemical potential is given instead of the number of particles . The probability of a particular micro-state is ${\displaystyle N}$ ${\displaystyle \mu }$ ${\displaystyle i}$

${\displaystyle p_{i}={\frac {1}{Z_{g}(\mu ,V,T)}}\mathrm {e} ^{-{\frac {E_{i}-\mu N_{i}}{k_{\mathrm {B} }T}}}.}$

The partition function is

${\displaystyle Z_{g}(\mu ,V,T)=\sum _{i}\mathrm {e} ^{-{\frac {E_{i}-\mu N_{i}}{k_{\mathrm {B} }T}}}.}$

In integral notation, the total state or the state integral is

${\displaystyle Z_{g}(\mu ,V,T)=\sum \limits _{N=0}^{\infty }\int \mathrm {e} ^{-{\frac {E(\mathbf {p,q} )-\mu N}{k_{\mathrm {B} }T}}}\,{\frac {\mathrm {d} \mathbf {p} \;\mathrm {d} \mathbf {q} }{h^{3N}N!}}.}$

The grand canonical partition function can be obtained from the canonical partition partition function and the fugacity : ${\displaystyle z=\exp(\mu /k_{\mathrm {B} }T)}$

${\displaystyle Z_{g}(\mu ,V,T)=\sum \limits _{N=0}^{\infty }Z_{k}(N,V,T)z^{N}=\sum _{N=0}^{\infty }Z_{k}(N,V,T)\,\mathrm {e} ^{\frac {\mu N}{k_{\mathrm {B} }T}}.}$

Calculation of the thermodynamic potentials

${\displaystyle {\begin{alignedat}{3}S(N,V,E)&=&&k_{\mathrm {B} }&&\,\ln Z_{m}(N,V,E)\\F(N,V,T)&=-&&k_{\mathrm {B} }T&&\,\ln Z_{k}(N,V,T)\\\Omega (\mu ,V,T)&=-&&k_{\mathrm {B} }T&&\,\ln Z_{g}(\mu ,V,T)\end{alignedat}}}$

Here is

• ${\displaystyle S}$the entropy
• ${\displaystyle F}$the free energy and
• ${\displaystyle \Omega }$the grand canonical potential .

See also

• Indistinguishable Particles

Individual evidence

1. ↑ Florian Scheck: Theoretical Physics 5: Statistical theory of heat . Springer, 2008, ISBN 978-3-540-79823-1 , pp. 98 ( online ).
2. ↑ P. Hertz , Ann. Phys. (Leipzig) 33, 225 (1910). P. Hertz , Ann. Phys. (Leipzig) 33, 537 (1910).
3. ↑ Wolfgang Nolting, Basic Course Theoretical Physics. Vol. 6: Statistical Physics, ISBN 978-3540205050 , p. 27

literature

• Richard Becker and Wolfgang Ludwig: Theory of Heat (Springer, Berlin, 1988), ISBN 3-540-15383-7
• Torsten Fließbach : Statistical Physics (1995), ISBN 3-86025-715-3 - An introduction to statistical physics and thermodynamics