NationMaster - Encyclopedia: Partition function (statistical mechanics)

In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... Partial plot of a function f. ... Fig. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Energy is an important concept in science, with its origins in physics via work, and it is very convenient quantity which thus finds applications throughout the natural sciences. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... This article assumes an understanding of algebra, analytic geometry, and the limit. ...

There are actually several different types of partition functions, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances. In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... The free energy is a measure of the amount of mechanical (or other) work that can be extracted from a system, and is helpful in engineering applications. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In physics, heat, symbolized by Q, is defined as transfer of thermal energy [1] Generally, heat is a form of energy transfer associated with the different motions of atoms, molecules and other particles that comprise matter when it is hot and when it is cold. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

1 Canonical partition function
2 Grand canonical partition function
3 References

Canonical partition function

Definition

Suppose we have a thermodynamically large system that is in constant thermal contact with the environment, which has temperature T, with both the volume of the system and the number of constituent particles fixed. This kind of system is called a canonical ensemble. Let us label the exact states (microstates) that the system can occupy by j (j = 1, 2, 3, ...), and denote the total energy of the system when it is in microstate j as E_j. Generally, these microstates can be regarded as discrete quantum states of the system. A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... A quantum state is any possible state in which a quantum mechanical system can be. ...

The canonical partition function is

$Z = sum_{j} e^{- beta E_j}$

where the "inverse temperature" β is conventionally defined as

$beta equiv frac{1}{k_BT}$

with k_B denoting Boltzmann's constant. Sometimes degeneracy of states is also used and the partition function becomes The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

$Z = sum_{j} g_jcdot e^{- beta E_j}$ ,

where $g j$ is the degeneracy factor.

In classical statistical mechanics, it is not really correct to express the partition function as a sum of discrete terms, as we have done. In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In this case, some form of coarse graining procedure must be carried out, which essentially amounts to treating two mechanical states as the same microstate if the differences in their position and momentum variables are "not too large". The partition function then takes the form of an integral. For instance, the partition function of a gas of N classical particles is Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ... Granularity is the extent to which a system contains discrete components of ever-smaller size. ... In calculus, the integral of a function is an extension of the concept of a sum. ...

$Z=frac{1}{N!h^{3N}} int , exp[-beta H(p_1 cdots p_N, x_1 cdots x_N)] ; mathrm{d}^3p_1 cdots mathrm{d}^3p_N , mathrm{d}^3x_1 cdots mathrm{d}^3x_N$

where h is some infinitesimal quantity with units of action (usually taken to be Planck's constant, in order to be consistent with quantum mechanics), and H is the classical Hamiltonian. The reason for the N! factor is discussed below. For simplicity, we will use the discrete form of the partition function in this article, but our results will apply equally well to the continuous form. In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Fig. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis): In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

$Z=operatorname{tr} ( e^{-beta H} )$

where H is the quantum Hamiltonian operator. The exponential of an operator can be defined, for purely physical considerations, using the exponential power series. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In mathematics, the exponential function can be characterized in many ways. ...

Meaning and significance

It may not be obvious why the partition function, as we have defined it above, is an important quantity. Firstly, let us consider what goes into it. The partition function is a function of the temperature T and the microstate energies E₁, E₂, E₃, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.

The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability P_j that the system occupies microstate j is

$P_j = frac{1}{Z} e^{- beta E_j}.$

This is the well-known Boltzmann factor. (For a detailed derivation of this result, see canonical ensemble.) The partition function thus plays the role of a normalizing constant (note that it does not depend on j), ensuring that the probabilities add up to one: In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...

$sum_j P_j = frac{1}{Z} sum_j e^{- beta E_j} = frac{1}{Z} Z = 1.$

This is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter Z stands for the German word Zustandssumme, "sum over states".

Calculating the thermodynamic total energy

In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value, or ensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities: In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In statistical mechanics, the ensemble average is defined as the weighted average of a molecular property of a system, over the set of states available to the system. ...

$langle E rangle = sum_j E_j P_j = frac{1}{Z} sum_j E_j e^{- beta E_j} = - frac{1}{Z} frac{partial}{partial beta} Z(beta, E_1, E_2, cdots) = - frac{partial ln Z}{partial beta}$

or, equivalently,

$langle Erangle = k_B T^2 frac{partial ln Z}{partial T}.$

Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner

$E_j = E_j^{(0)} + lambda A_j qquad mbox{for all}; j$

then the expected value of A is

$langle Arangle = sum_j A_j P_j = -frac{1}{beta} frac{partial}{partiallambda} ln Z(beta,lambda).$

This provides us with a trick for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set λ to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory. In theoretical physics, a source field is a field whose multiple appears in the action, multiplied by the original field . ... This article or section is in need of attention from an expert on the subject. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Relation to thermodynamic variables

In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.

As we have already seen, the thermodynamic energy is

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

The variance in the energy (or "energy fluctuation") is In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

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

The heat capacity is To meet Wikipedias quality standards, this article or section may require cleanup. ...

$C_v = frac{partial langle Erangle}{partial T} = frac{1}{k_B T^2} langle delta E^2 rangle.$

The entropy is Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

$S equiv -k_Bsum_j P_jln P_j= k_B (ln Z + beta langle Erangle)=frac{partial}{partial T}(k_B T ln Z) =-frac{partial A}{partial T}$

where A is the Helmholtz free energy defined as A = U - TS, where U=<E> is the total energy and S is the entropy, so that In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the â€œusefulâ€ work obtainable from a closed thermodynamic system at a constant temperature. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ...

$A = langle Erangle -TS=- k_B T ln Z.$

Partition functions of subsystems

Suppose a system is subdivided into N sub-systems with negligible interaction energy. If the partition functions of the sub-systems are ζ₁, ζ₂, ..., ζ_N, then the partition function of the entire system is the product of the individual partition functions:

$Z =prod_{j=1}^{N} zeta_j.$

If the sub-systems have the same physical properties, then their partition functions are equal, ζ₁ = ζ₂ = ... = ζ, in which case

Z = ζ N .

However, there is a well-known exception to this rule. If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a N! (N factorial): Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... Fig. ... For factorial rings in mathematics, see unique factorisation domain. ...

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

This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox. The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

Examples

A specific example of the partition function, expressed in terms of the mathematical formalism of measure theory, is presented in the article on the Potts model. In mathematics, a measure is a function that assigns a number, e. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ...

Grand canonical partition function

Definition

In a manner similar to the definition of the canonical partition function for the canonical ensemble, we can define a grand canonical partition function for a grand canonical ensemble, a system that can exchange both heat and particles with the environment, which has a constant temperature T, volume V, and chemical potential μ. The grand canonical partition function, although conceptually more involved, simplifies the calculation of the physics of quantum systems. The grand canonical partition function $mathcal{Z}$ for an ideal quantum gas is written: A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ... In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

$mathcal{Z} = sum_{N=0}^infty,sum_{{n_i}},prod_i e^{-beta n_i(epsilon_i-mu)}$

where N is the total number of particles in the volume V, and index i runs over every microstate of the system, with n_i being the number of particles in state i and ε_i being the energy of state i. {n_i} is the set of all possible occupation numbers for each of the microstates such that Σ_in_i = N.

For example, consider the N = 3 term in the above sum. One possible set of occupation numbers would be {n_i} = 0,1,0,2,0... and the contribution of this set of occupation numbers to the N = 3 term would be

$prod_i e^{-beta n_i(epsilon_i-mu)}=e^{-beta(epsilon_1-mu)},e^{-2beta(epsilon_3-mu)}.$

For bosons, the occupation numbers can take any integer values as long as their sum is equal to N. For fermions, the Pauli exclusion principle requires that the occupation numbers only be 0 or 1, again adding up to N. Boson (game) Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...

Specific expressions

The above expression for the grand partition function can be shown to be mathematically equivalent to:

$mathcal{Z} = prod_i mathcal{Z}_i.$

(Note: the above product is sometimes taken over all states with equal energy, rather than over each state, in which case the individual partition functions must be raised to a power g_i where g_i is the number of such states. g_i is also referred to as the "degeneracy" of states.)

For a system composed of bosons: In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ...

$mathcal{Z}_i = sum_{n_i=0}^infty e^{-beta n_i(epsilon_i-mu)} = frac{1}{1-e^{-beta (epsilon_i-mu)}}$

and for a system composed of fermions: In particle physics, fermions are particles with half-integer spin. ...

$mathcal{Z}_i = sum_{n_i=0}^1 e^{-beta n_i(epsilon_i-mu)} = left(1+e^{-beta (epsilon_i-mu)}right).$

For the case of a Maxwell-Boltzmann gas, we must use "correct Boltzmann counting" and divide the Boltzmann factor $e^{-beta (epsilon_i-mu)}$ by n_i!.

$mathcal{Z}_i = sum_{n_i=0}^infty frac{e^{-beta n_i(epsilon_i-mu)}}{n_i!} = exp left( e^{-beta (epsilon_i-mu)}right).$

Relation to thermodynamic variables

Just as with the canonical partition function, the grand canonical partition function can be used to calculate thermodynamic and statistical variables of the system. As with the canonical ensemble, the thermodynamic quantities are not fixed, but have a statistical distribution about a mean or expected value. Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ...

Defining α=-βμ, the most probable occupation numbers are:

$langle n_irangle = -left(frac{partialln(mathcal{Z}_i)}{partial alpha} right)_{beta,V} = frac{1}{beta}left(frac{partialln(mathcal{Z}_i)}{partial mu} right)_{beta,V}.$

For Boltzmann particles this yields:

$langle n_irangle = e^{-beta(epsilon_i-mu)}.$

For bosons:

$langle n_irangle = frac{1}{e^{beta(epsilon_i-mu)}-1}.$

For fermions:

$langle n_irangle = frac{1}{e^{beta(epsilon_i-mu)}+1}.$

which are just the results found using the canonical ensemble for Maxwell-Boltzmann statistics, Bose-Einstein statistics and Fermi-Dirac statistics, respectively. (Note: the degeneracy g_i is missing from the above equations because the index i is summing over individual microstates rather than energy eigenvalues.)
Total number of particles

$langle Nrangle = -left(frac{partialln(mathcal{Z})}{partial alpha} right)_{beta,V} = frac{1}{beta}left(frac{partialln(mathcal{Z})}{partial mu} right)_{beta,V}.$
Variance in total number of particles

$langle (delta N)^2 rangle = left(frac{partial^2ln(mathcal{Z})}{partial alpha^2} right)_{beta,V}.$
Internal energy

$langle Erangle = -left(frac{partial ln(mathcal{Z})}{partial beta} right)_{mu,V} + mu langle Nrangle.$
Variance in internal energy

$langle (delta E)^2 rangle = left(frac{partial^2 ln(mathcal{Z})}{partial beta^2} right)_{mu,V}.$
Pressure

$langle Prangle=frac{1}{beta}left(frac{partialln(mathcal{Z})}{partial V} right)_{mu,beta}.$
Mechanical equation of state

$langle PVrangle=frac{ln(mathcal{Z})}{beta}.$

It has been suggested that the section Physical applications of the Maxwell-Boltzmann distribution from the article Maxwell-Boltzmann distribution be merged into this article or section. ... In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... Fermi-Dirac distribution as a function of Îµ/Î¼ plotted for 4 different temperatures. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... In physics and thermodynamics, an equation of state is a constitutive equation describing the state of matter under a given set of physical conditions. ...

Relation to potential V

For the case of a non-interacting gas, using the "Semiclassical Approach" we can write (approximately) the inverse of the potential in the form:

$frac{1}{2ipi}int_{c-iinfty}^{c+iinfty}mathrm{d}sfrac{Z(s)}{sqrt (pi s)}e^{st} sim V^{-1} (t)$

$s= frac{1}{k_{B} T}$ (valid for high T )

supposing that the Hamiltonian of every particle is H=T+V .

Discussion

Before specific results can be obtained from the grand canonical partition function, the energy levels of the system under consideration need to be specified. For example, the particle in a box model or particle in a harmonic oscillator well provide a particular set of energy levels and are a convenient way to discuss the properties of a quantum fluid. (See the gas in a box and gas in a harmonic trap articles for a description of quantum fluids.) In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with... In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ... The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ... The results of the quantum harmonic oscillator can be used to look at the equilibrium situation for a quantum ideal gas in a harmonic trap which is a harmonic potential containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

These results may be used to construct the grand partition function to describe an ideal Bose gas or Fermi gas, and can be used as well to describe a classical ideal gas. An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ... A Fermi gas is a collection of non-interacting fermions. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

References

Huang, Kerson, "Statistical Mechanics", John Wiley & Sons, New York, 1967.
A. Isihara, "Statistical Physics", Academic Press, New York, 1971.
Kelly, James J, (Lecture notes)
L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996.

Categories: Fundamental physics concepts | Statistical mechanics

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Contents

Canonical partition function GA_googleFillSlot("encyclopedia_square");

Definition

Meaning and significance

Calculating the thermodynamic total energy

Relation to thermodynamic variables

Partition functions of subsystems

Examples

Grand canonical partition function

Definition

Specific expressions

Relation to thermodynamic variables

Relation to potential V

Discussion

References

Canonical partition function