Statistical amount

Statistical amount (or part amount ) (indicated by $Z$ ${\ displaystyle Z}$ $Z$ from him. Zustandssumme (sum over states) is an important quantity in statistical physics , containing information on the statistical properties of a system in a state of thermodynamic equilibrium . It is a function of temperature and other parameters, such as volume . Many thermodynamic quantities of a system, such as energy , free energy , entropy, and pressure , can be expressed in terms of the partition function and its derivatives .

There are several types of statistical sums, each of which corresponds to different statistical ensembles . The canonical statistical sum refers to the canonical statistical ensemble in which the system can exchange heat with the environment at a fixed temperature, volume and number of particles. A large canonical statistical sum refers to a large canonical statistical ensemble in which the system can exchange both heat and particles with the environment at a fixed temperature, volume, and chemical potential . In other situations, you can define other types of statistical sums.

Content

The statistical sum in the canonical ensemble

Definition

Suppose that there is a system obeying the laws of thermodynamics that is in constant thermal contact with a medium that has a temperature $T$ ${\ displaystyle T}$ , and the volume of the system and the number of its constituent particles are fixed. In such a situation, the system belongs to the canonical ensemble . We denote the exact states in which the system can be located by $j$ ${\ displaystyle j}$ $(j=1,2,3,\ldots )$ ${\ displaystyle (j = 1,2,3, \ ldots)}$ , and the total energy of the system is able to $j$ ${\ displaystyle j}$ - $E_{j}$ ${\ displaystyle E_ {j}}$ . As a rule, these microstates can be considered as discrete quantum states of the system.

The canonical partition function is

Z=\sum _{j}e^{-\beta E_{j}},

{\ displaystyle Z = \ sum _ {j} e ^ {- \ beta E_ {j}},}

where is the return temperature $\beta$ ${\ displaystyle \ beta}$ defined as

\beta \equiv {\frac {1}{k_{B}T}},

{\ displaystyle \ beta \ equiv {\ frac {1} {k_ {B} T}},}

but $k_{B}$ ${\ displaystyle k_ {B}}$ Is the Boltzmann constant . In classical statistical mechanics, it would be incorrect to determine the partition function as the sum of discrete terms, as in the above formula. In classical mechanics, the coordinates and momenta of particles can change continuously, and many microstates are uncountable . In this case, it is necessary to divide the phase space into cells, that is, two microstates are considered identical if their differences in coordinates and momenta are not too large. In this case, the partition function takes the form of an integral . For example, the statistical amount of gas from $N$ ${\ displaystyle N}$ classical particles equal

Z={\frac {1}{N!h^{3N}}}\int \exp[-\beta H(p_{1},\ldots ,p_{N},x_{1},\ldots ,x_{N})]\,d^{3}p_{1}\ldots d^{3}p_{N}\,d^{3}x_{1}\ldots d^{3}x_{N},

{\ displaystyle Z = {\ frac {1} {N! h ^ {3N}}} \ int \ exp [- \ beta H (p_ {1}, \ ldots, p_ {N}, x_ {1}, \ ldots, x_ {N})] \, d ^ {3} p_ {1} \ ldots d ^ {3} p_ {N} \, d ^ {3} x_ {1} \ ldots d ^ {3} x_ { N},}

Where $h$ ${\ displaystyle h}$ Is a certain dimension of the action (which must be equal to the Planck constant to correspond to quantum mechanics ), and $H$ ${\ displaystyle H}$ - classical Hamiltonian . Reasons for the multiplier $N!$ ${\ displaystyle N!}$ explained below . For simplicity, this article will use the discrete form of the partition function, but the results obtained apply equally to the continuous form.

In quantum mechanics, a partition function can be written more formally as a trace in the state space (which does not depend on the choice of basis ):

Z=\mathrm {tr} \,(e^{-\beta H}),

{\ displaystyle Z = \ mathrm {tr} \, (e ^ {- \ beta H}),}

Where $H$ ${\ displaystyle H}$ - Hamilton operator . The exponent of the operator is determined by expanding in a power series .

Meaning and Significance

First, consider what it depends on. The statistical sum is a function, first of all, of temperature $T$ ${\ displaystyle T}$ and in the second - microstate energies $E_{1},E_{2},E_{3}$ ${\ displaystyle E_ {1}, E_ {2}, E_ {3}}$ etc. The energies of microstates are determined by other thermodynamic quantities, such as the number of particles and volume, as well as microscopic properties, such as the mass of particles. This dependence on microscopic properties is fundamental in statistical mechanics. Using the model of microscopic components of the system, it is possible to calculate the energies of microstates, and hence the statistical sum, which allows one to calculate all other thermodynamic properties of the system.

A statistical sum can be used to calculate thermodynamic quantities, since it has a very important statistical meaning. Probability $P_{j}$ ${\ displaystyle P_ {j}}$ with which the system is in a microstate $j$ ${\ displaystyle j}$ is equal to

P_{j}={\frac {1}{Z}}e^{-\beta E_{j}}.

{\ displaystyle P_ {j} = {\ frac {1} {Z}} e ^ {- \ beta E_ {j}}.}

The statistical sum is included in the Gibbs distribution in the form of a normalization factor (it does not depend on $j$ ${\ displaystyle j}$ ), ensuring equality to the unit of the sum of the probabilities:

\sum _{j}P_{j}={\frac {1}{Z}}\sum _{j}e^{-\beta E_{j}}={\frac {1}{Z}}Z=1.

{\ displaystyle \ sum _ {j} P_ {j} = {\ frac {1} {Z}} \ sum _ {j} e ^ {- \ beta E_ {j}} = {\ frac {1} {Z }} Z = 1.}

Thermodynamic Total Energy Calculation

To demonstrate the usefulness of the partition function, we calculate the thermodynamic value of the total energy. This is just a mathematical expectation , or the ensemble average value of energy equal to the sum of the energies of microstates taken with weights equal to their probabilities:

\langle E\rangle =\sum _{j}E_{j}P_{j}={\frac {1}{Z}}\sum _{j}E_{j}e^{-\beta E_{j}}=-{\frac {1}{Z}}{\frac {\partial }{\partial \beta }}Z(\beta ,\;E_{1},\;E_{2},\;\ldots )=-{\frac {\partial \ln Z}{\partial \beta }}

{\ displaystyle \ langle E \ rangle = \ sum _ {j} E_ {j} P_ {j} = {\ frac {1} {Z}} \ sum _ {j} E_ {j} e ^ {- \ beta E_ {j}} = - {\ frac {1} {Z}} {\ frac {\ partial} {\ partial \ beta}} Z (\ beta, \; E_ {1}, \; E_ {2}, \; \ ldots) = - {\ frac {\ partial \ ln Z} {\ partial \ beta}}}

or what is the same

\langle E\rangle =k_{B}T^{2}{\frac {\partial \ln Z}{\partial T}}.

{\ displaystyle \ langle E \ rangle = k_ {B} T ^ {2} {\ frac {\ partial \ ln Z} {\ partial T}}.}

You can also notice that if the energies of microstates depend on the parameter $\lambda$ ${\ displaystyle \ lambda}$ as

E_{j}=E_{j}^{(0)}+\lambda A_{j}

{\ displaystyle E_ {j} = E_ {j} ^ {(0)} + \ lambda A_ {j}}

for all $j$ ${\ displaystyle j}$ , then the average value $A$ ${\ displaystyle A}$ equally

\langle A\rangle =\sum _{j}A_{j}P_{j}=-{\frac {1}{\beta }}{\frac {\partial }{\partial \lambda }}\ln Z(\beta ,\;\lambda ).

{\ displaystyle \ langle A \ rangle = \ sum _ {j} A_ {j} P_ {j} = - {\ frac {1} {\ beta}} {\ frac {\ partial} {\ partial \ lambda}} \ ln Z (\ beta, \; \ lambda).}

This is the basis of the technique, which allows calculating the average values of many microscopic quantities. It is necessary to artificially add this quantity to the energy of microstates (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new statistical sum and average value, and then put in the final expression $\lambda$ ${\ displaystyle \ lambda}$ equal to zero. A similar method is used in quantum field theory .

Relationship with thermodynamic quantities

This section describes the relationship of the partition function with various thermodynamic parameters of the system. These results can be obtained using the method described in the previous section and various thermodynamic relations.

As we have seen, energy is equal to

\langle E\rangle =-{\frac {\partial \ln Z}{\partial \beta }}.

{\ displaystyle \ langle E \ rangle = - {\ frac {\ partial \ ln Z} {\ partial \ beta}}.}

The fluctuation of energy is equal to

\langle \delta E^{2}\rangle \equiv \langle (E-\langle E\rangle )^{2}\rangle ={\frac {\partial ^{2}\ln Z}{\partial \beta ^{2}}}.

{\ displaystyle \ langle \ delta E ^ {2} \ rangle \ equiv \ langle (E- \ langle E \ rangle) ^ {2} \ rangle = {\ frac {\ partial ^ {2} \ ln Z} {\ partial \ beta ^ {2}}}.}

Heat capacity is equal

c_{v}={\frac {\partial \langle E\rangle }{\partial T}}={\frac {1}{k_{B}T^{2}}}\langle \delta E^{2}\rangle .

{\ displaystyle c_ {v} = {\ frac {\ partial \ langle E \ rangle} {\ partial T}} = {\ frac {1} {k_ {B} T ^ {2}}} \ langle \ delta E ^ {2} \ rangle.}

Entropy is equal to

S\equiv -k_{B}\sum _{j}P_{j}\ln P_{j}=k_{B}(\ln Z+\beta \langle E\rangle )={\frac {\partial }{\partial T}}(k_{B}T\ln Z)=-{\frac {\partial F}{\partial T}},

{\ displaystyle S \ equiv -k_ {B} \ sum _ {j} P_ {j} \ ln P_ {j} = k_ {B} (\ ln Z + \ beta \ langle E \ rangle) = {\ frac {\ partial} {\ partial T}} (k_ {B} T \ ln Z) = - {\ frac {\ partial F} {\ partial T}},}

Where $F$ ${\ displaystyle F}$ - free energy , defined as $F=E-TS$ ${\ displaystyle F = E-TS}$ where $E$ ${\ displaystyle E}$ Is the total energy, and $S$ ${\ displaystyle S}$ Is entropy , so

F=\langle E\rangle -TS=-k_{B}T\ln Z.

{\ displaystyle F = \ langle E \ rangle -TS = -k_ {B} T \ ln Z.}

Statistical sum of subsystems

Assume that the system consists of $N$ ${\ displaystyle N}$ subsystems, the interaction between which is negligible. If the statistical sums of the subsystems are equal $\zeta _{1},\;\zeta _{2},\;\ldots ,\;\zeta _{N}$ ${\ displaystyle \ zeta _ {1}, \; \ zeta _ {2}, \; \ ldots, \; \ zeta _ {N}}$ , then the statistical sum of the whole system is equal to the product of individual statistical sums:

Z=\prod _{j=1}^{N}\zeta _{j}.

{\ displaystyle Z = \ prod _ {j = 1} ^ {N} \ zeta _ {j}.}

If the subsystems have the same physical properties, then their statistical sums are the same: $\zeta _{1}=\zeta _{2}=\ldots =\zeta$ ${\ displaystyle \ zeta _ {1} = \ zeta _ {2} = \ ldots = \ zeta}$ , and in this case

Z=\zeta ^{N}.

{\ displaystyle Z = \ zeta ^ {N}.}

There is, however, one known exception to this rule. If the subsystems are identical particles , that is, based on the principles of quantum mechanics, they cannot be distinguished even in principle, the total statistical sum should be divided into $N!$ ${\ displaystyle N!}$ :

Z={\frac {\zeta ^{N}}{N!}}.

{\ displaystyle Z = {\ frac {\ zeta ^ {N}} {N!}}.}

This is done so as not to take into account the same microstate several times.

The statistical sum of the grand canonical ensemble

Definition

Similarly to the canonical partition function for a canonical ensemble , one can define a large canonical partition sum for a large canonical ensemble - a system that can exchange with medium and heat, and particles, and has a constant temperature $T$ ${\ displaystyle T}$ volume $V$ ${\ displaystyle V}$ and chemical potential $\mu$ ${\ displaystyle \ mu}$ . A large canonical partition function, although more difficult to understand, simplifies the calculation of quantum systems. Large canonical partition function ${\mathcal {Z}}$ ${\ displaystyle {\ mathcal {Z}}}$ for a quantum ideal gas is written as:

{\mathcal {Z}}=\sum _{N=0}^{\infty }\,\sum _{\{n_{i}\}}\,\prod _{i}e^{-\beta n_{i}(\varepsilon _{i}-\mu )},

{\ displaystyle {\ mathcal {Z}} = \ sum _ {N = 0} ^ {\ infty} \, \ sum _ {\ {n_ {i} \}} \, \ prod _ {i} e ^ { - \ beta n_ {i} (\ varepsilon _ {i} - \ mu)},}

Where $N$ ${\ displaystyle N}$ - total number of particles in volume $V$ ${\ displaystyle V}$ index $i$ ${\ displaystyle i}$ runs through all the microstates of the system, $n_{i}$ ${\ displaystyle n_ {i}}$ - the number of particles in the state $i$ ${\ displaystyle i}$ , but $\varepsilon _{i}$ ${\ displaystyle \ varepsilon _ {i}}$ - energy is able $i$ ${\ displaystyle i}$ . $\{n_{i}\}$ ${\ displaystyle \ {n_ {i} \}}$ - all kinds of sets of fill numbers for each microstate, such that $\sum _{i}n_{i}=N$ ${\ displaystyle \ sum _ {i} n_ {i} = N}$ . Consider, for example, the term corresponding to $N=3$ ${\ displaystyle N = 3}$ . One possible set of fill numbers would be $\{n_{i}\}=0,\;1,\;0,\;2,\;0,\ldots$ ${\ displaystyle \ {n_ {i} \} = 0, \; 1, \; 0, \; 2, \; 0, \ ldots}$ , it contributes to the term with $N=3$ ${\ displaystyle N = 3}$ equal to

\prod _{i}e^{-\beta n_{i}(\varepsilon _{i}-\mu )}=e^{-\beta (\varepsilon _{1}-\mu )}\,e^{-2\beta (\varepsilon _{3}-\mu )}.

{\ displaystyle \ prod _ {i} e ^ {- \ beta n_ {i} (\ varepsilon _ {i} - \ mu)} = e ^ {- \ beta (\ varepsilon _ {1} - \ mu)} \, e ^ {- 2 \ beta (\ varepsilon _ {3} - \ mu)}.}

For bosons , the occupation numbers can take any non-negative integer values despite the fact that their sum is $N$ ${\ displaystyle N}$ . For fermions , in accordance with the Pauli prohibition principle , the occupation numbers can only be 0 or 1, but their sum is again equal to $N$ ${\ displaystyle N}$ .

Special cases

It can be shown that the indicated expression for a large canonical partition function is mathematically equivalent to the following:

{\mathcal {Z}}=\prod _{i}{\mathcal {Z}}_{i}.

{\ displaystyle {\ mathcal {Z}} = \ prod _ {i} {\ mathcal {Z}} _ {i}.}

(This product is sometimes taken for all energy values, and not for individual states, in which case each individual statistical sum should be raised to a power $g_{i}$ ${\ displaystyle g_ {i}}$ where $g_{i}$ ${\ displaystyle g_ {i}}$ - the number of states with such energy. $g_{i}$ ${\ displaystyle g_ {i}}$ also called the degree of degeneration.)

For a system consisting of bosons :

{\mathcal {Z}}_{i}=\sum _{n_{i}=0}^{\infty }e^{-\beta n_{i}(\varepsilon _{i}-\mu )}={\frac {1}{1-e^{-\beta (\varepsilon _{i}-\mu )}}},

{\ displaystyle {\ mathcal {Z}} _ {i} = \ sum _ {n_ {i} = 0} ^ {\ infty} e ^ {- \ beta n_ {i} (\ varepsilon _ {i} - \ mu)} = {\ frac {1} {1-e ^ {- \ beta (\ varepsilon _ {i} - \ mu)}}},}

and for a system consisting of fermions :

{\mathcal {Z}}_{i}=\sum _{n_{i}=0}^{1}e^{-\beta n_{i}(\varepsilon _{i}-\mu )}=1+e^{-\beta (\varepsilon _{i}-\mu )}.

{\ displaystyle {\ mathcal {Z}} _ {i} = \ sum _ {n_ {i} = 0} ^ {1} e ^ {- \ beta n_ {i} (\ varepsilon _ {i} - \ mu )} = 1 + e ^ {- \ beta (\ varepsilon _ {i} - \ mu)}.}

In the case of the Maxwell-Boltzmann gas, it is necessary to correctly calculate the states and divide the Boltzmann factor $e^{-\beta (\varepsilon _{i}-\mu )}$ ${\ displaystyle e ^ {- \ beta (\ varepsilon _ {i} - \ mu)}}$ on $n_{i}!$ ${\ displaystyle n_ {i}!}$

{\mathcal {Z}}_{i}=\sum _{n_{i}=0}^{\infty }{\frac {e^{-\beta n_{i}(\varepsilon _{i}-\mu )}}{n_{i}!}}=\exp \left(e^{-\beta (\varepsilon _{i}-\mu )}\right).

{\ displaystyle {\ mathcal {Z}} _ {i} = \ sum _ {n_ {i} = 0} ^ {\ infty} {\ frac {e ^ {- \ beta n_ {i} (\ varepsilon _ { i} - \ mu)}} {n_ {i}!}} = \ exp \ left (e ^ {- \ beta (\ varepsilon _ {i} - \ mu)} \ right).}

Relationship with thermodynamic quantities

Like the canonical partition function, the large canonical partition function can be used to calculate the thermodynamic and statistical quantities of the system. As in the canonical ensemble, thermodynamic quantities are not fixed, but statistically distributed around the average value. Marking $\alpha =-\beta \mu$ ${\ displaystyle \ alpha = - \ beta \ mu}$ , we get the average values of the fill numbers:

\langle n_{i}\rangle =-\left({\frac {\partial \ln {\mathcal {Z}}_{i}}{\partial \alpha }}\right)_{\beta ,\;V}={\frac {1}{\beta }}\left({\frac {\partial \ln {\mathcal {Z}}_{i}}{\partial \mu }}\right)_{\beta ,\;V}.

{\ displaystyle \ langle n_ {i} \ rangle = - \ left ({\ frac {\ partial \ ln {\ mathcal {Z}} _ {i}} {\ partial \ alpha}} \ right) _ {\ beta , \; V} = {\ frac {1} {\ beta}} \ left ({\ frac {\ partial \ ln {\ mathcal {Z}} _ {i}} {\ partial \ mu}} \ right) _ {\ beta, \; V}.}

For Boltzmann particles, this gives:

\langle n_{i}\rangle =e^{-\beta (\varepsilon _{i}-\mu )}.

{\ displaystyle \ langle n_ {i} \ rangle = e ^ {- \ beta (\ varepsilon _ {i} - \ mu)}.}

For bosons:

\langle n_{i}\rangle ={\frac {1}{e^{\beta (\varepsilon _{i}-\mu )}-1}}.

{\ displaystyle \ langle n_ {i} \ rangle = {\ frac {1} {e ^ {\ beta (\ varepsilon _ {i} - \ mu)} - 1}}.}

For fermions:

\langle n_{i}\rangle ={\frac {1}{e^{\beta (\varepsilon _{i}-\mu )}+1}},

{\ displaystyle \ langle n_ {i} \ rangle = {\ frac {1} {e ^ {\ beta (\ varepsilon _ {i} - \ mu)} + 1}},}

which coincides with the results obtained using the canonical ensemble for Maxwell – Boltzmann statistics, Bose – Einstein statistics, and Fermi – Dirac statistics, respectively. (Degree of degeneration $g_{i}$ ${\ displaystyle g_ {i}}$ missing in these equations because the index $i$ ${\ displaystyle i}$ numbers individual states, not energy levels.)

Total number of particles

\langle N\rangle =-\left({\frac {\partial \ln {\mathcal {Z}}}{\partial \alpha }}\right)_{\beta ,\;V}={\frac {1}{\beta }}\left({\frac {\partial \ln {\mathcal {Z}}}{\partial \mu }}\right)_{\beta ,\;V}.

{\ displaystyle \ langle N \ rangle = - \ left ({\ frac {\ partial \ ln {\ mathcal {Z}}} {\ partial \ alpha}} \ right) _ {\ beta, \; V} = { \ frac {1} {\ beta}} \ left ({\ frac {\ partial \ ln {\ mathcal {Z}}} {\ partial \ mu}} \ right) _ {\ beta, \; V}.}

Fluctuation of the total number of particles

\mathrm {var} \,(N)=\left({\frac {\partial ^{2}\ln {\mathcal {Z}}}{\partial \alpha ^{2}}}\right)_{\beta ,\;V}.

{\ displaystyle \ mathrm {var} \, (N) = \ left ({\ frac {\ partial ^ {2} \ ln {\ mathcal {Z}}} {\ partial \ alpha ^ {2}}} \ right ) _ {\ beta, \; V}.}

Internal energy

\langle E\rangle =-\left({\frac {\partial \ln {\mathcal {Z}}}{\partial \beta }}\right)_{\mu ,\;V}+\mu \langle N\rangle .

{\ displaystyle \ langle E \ rangle = - \ left ({\ frac {\ partial \ ln {\ mathcal {Z}}} {\ partial \ beta}} \ right) _ {\ mu, \; V} + \ mu \ langle N \ rangle.}

Fluctuation of internal energy

\mathrm {var} \,(E)=\left({\frac {\partial ^{2}\ln {\mathcal {Z}}}{\partial \beta ^{2}}}\right)_{\mu ,\;V}.

{\ displaystyle \ mathrm {var} \, (E) = \ left ({\ frac {\ partial ^ {2} \ ln {\ mathcal {Z}}} {\ partial \ beta ^ {2}}} \ right ) _ {\ mu, \; V}.}

Pressure

\langle P\rangle ={\frac {1}{\beta }}\left({\frac {\partial \ln {\mathcal {Z}}}{\partial V}}\right)_{\mu ,\;\beta }.

{\ displaystyle \ langle P \ rangle = {\ frac {1} {\ beta}} \ left ({\ frac {\ partial \ ln {\ mathcal {Z}}} {\ partial V}} \ right) _ { \ mu, \; \ beta}.}

Mechanical equation of state

\langle PV\rangle ={\frac {\ln {\mathcal {Z}}}{\beta }}.

{\ displaystyle \ langle PV \ rangle = {\ frac {\ ln {\ mathcal {Z}}} {\ beta}}.}

Literature

Kubo R. Statistical Mechanics. - M .: Mir, 1967.
Huang K. Statistical Mechanics. - M .: Mir, 1966. (Huang, Kerson, "Statistical Mechanics", John Wiley & Sons, New York, 1967.)
Ishihara A. Statistical Physics. - M.: Mir, 1973. (Isihara A. “Statistical Physics.” - New York: Academic Press, 1971.)
Kelly, James J. Lecture notes .
Landau L.D. , Lifshits E.M. Statistical Physics. Part 1. - Edition 5th. - M .: Fizmatlit , 2005 .-- 616 p. - (“ Theoretical Physics ”, Volume V). - ISBN 5-9221-0054-8 . .

Statistical amount

Content

The statistical sum in the canonical ensemble

Definition

Meaning and Significance

Thermodynamic Total Energy Calculation

Relationship with thermodynamic quantities

Statistical sum of subsystems

The statistical sum of the grand canonical ensemble

Definition

Special cases

Relationship with thermodynamic quantities

Literature

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