Partition function (statistical mechanics)

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Partition function (statistical mechanics)

In statistical mechanics

Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics 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....
, the partition function Z is an important quantity that encodes the statistical

Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
properties of a system in thermodynamic equilibrium

Thermodynamic equilibrium

In thermodynamics, a thermodynamics#Thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium....
. It is a function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
of temperature

Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
and other parameters, such as the volume

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
enclosing a gas. Most of the aggregate thermodynamic

Thermodynamics

In physics, thermodynamics is the study of the conversion of heat energy into different forms of energy ; different energy conversions into heat energy; and its relation to macroscopic variables such as temperature, pressure, and volume....
variables of the system, such as the total energy

Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
, free energy

Thermodynamic free energy

In thermodynamics, the term thermodynamic free energy refers to the amount of Work that can be extracted from a system, and is helpful in engineering applications....
, entropy

Entropy

In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....
, and pressure

Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
, can be expressed in terms of the partition function or its derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
s.

There are actually several different types of partition functions, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy

Thermodynamic free energy

In thermodynamics, the term thermodynamic free energy refers to the amount of Work that can be extracted from a system, and is helpful in engineering applications....
.) The canonical partition function applies to a canonical ensemble

Canonical ensemble

A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system....
, in which the system is allowed to exchange heat

Heat

In physics and thermodynamics, heat is any transfer of energy from one body or thermodynamic system to another due to a difference in temperature....
with the environment at fixed temperature, volume, and number of particles.

Discussion

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Encyclopedia

In statistical mechanics

Statistical mechanics

Statistics

Thermodynamic equilibrium

Function (mathematics)

Temperature

Volume

Thermodynamics

Energy

Thermodynamic free energy

In thermodynamics, the term thermodynamic free energy refers to the amount of Work that can be extracted from a system, and is helpful in engineering applications....
, entropy

Entropy

Pressure

Derivative

Thermodynamic free energy

Canonical ensemble

Heat

Grand canonical ensemble

In statistical mechanics, the grand canonical ensemble is a statistical ensemble , where each system is in equilibrium with an external reservoir with respect to both particle and energy exchange....
, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential

Chemical potential

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

Canonical ensemble

A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system....
. Let us label with s (s = 1, 2, 3, ...) the exact states (microstates) that the system can occupy, and denote the total energy of the system when it is in microstate s as E_s. Generally, these microstates can be regarded as discrete quantum state

Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
s of the system.

The canonical partition function is

where the "inverse temperature" β

Thermodynamic beta

In statistical mechanics, the thermodynamic beta is a numerical quantity related to the thermodynamic temperature of a system. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and thermodynamics....
is conventionally defined as

with k_B denoting Boltzmann's constant. Because the energy levels of a system are often degenerate, we can write the partition function in terms of energy levels (indexed by j) as follows:

where 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

Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable

Uncountable set

In mathematics, an uncountable set is an infinite Set which is too big to be countable set. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the natural numbers....
. 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

Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
. For instance, the partition function of a gas of N classical particles is

indicate particle momenta

indicate particle positions

is a shorthand notation serving as a reminder that the and are vectors in three dimensional space

where h is some infinitesimal quantity with units of action

Action (physics)

In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the Phase of the wave function has changed....
(usually taken to be Planck's constant, in order to be consistent with quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
), and H is the classical Hamiltonian

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
. 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. For more details on the derivation of the above classical partition function, see , where the partition function is denoted by , instead of , which is used to denote a different quantity called .

In quantum mechanics, the partition function can be more formally written as a trace

Trace (linear algebra)

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 of A, i.e.,...
over the state space

Mathematical formulation of quantum mechanics

The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics....
(which is independent of the choice of basis

Basis (linear algebra)

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others....
):

where H is the quantum Hamiltonian operator

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the Spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system....
. The exponential of an operator can be defined, for purely physical considerations, using the exponential power series

Characterizations of the exponential function

In mathematics, the exponential function can be characterization in many ways. The following characterizations are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other....
.

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_s that the system occupies microstate s is

is the well-known Boltzmann factor

Boltzmann factor

In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a state in a multi-state system in thermodynamic equilibrium at temperature ....
. (For a detailed derivation of this result, see canonical ensemble

Canonical ensemble

A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system....
.) The partition function thus plays the role of a normalizing constant (note that it does not depend on s), ensuring that the probabilities add up to one:

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

German language

German is a West Germanic languages, thus related to and classified alongside English language and Dutch language. It is one of the world's world language and the most widely spoken mother tongue in the European Union....
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

Expected value

In probability theory and statistics, the expected value of a random variable is the Lebesgue integral of the random variable with respect to its probability measure....
, or ensemble average

Ensemble average

In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the micro-state of a system , according to the distribution of the system on its micro-states in this statistical mechanics....
for the energy, which is the sum of the microstate energies weighted by their probabilities:

or, equivalently,

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

then the expected value of A is

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 ? to zero in the final expression. This is analogous to the source field

Source field

In theoretical physics, a source field is a field whose multipleappears in the action, multiplied by the original field . Consequently, the source field appears on the right-hand side of the equations of motion for ....
method used in the path integral formulation

Path integral formulation

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a probability amplitude....
of quantum field theory

Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
.

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

The variance

Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
in the energy (or "energy fluctuation") is

The heat capacity is

The entropy

Entropy

where A is the Helmholtz free energy

Helmholtz free energy

In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the ?useful? work obtainable from a closed system thermodynamic thermodynamic system at a constant temperature and volume....
defined as A = U - TS, where U= is the total energy and S is the entropy

Entropy

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 ?₁, ?₂, ..., ?_N, then the partition function of the entire system is the product of the individual partition functions:

If the sub-systems have the same physical properties, then their partition functions are equal, ?₁ = ?₂ = ... = ?, in which case

However, there is a well-known exception to this rule. If the sub-systems are actually identical particles

Identical particles

Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, as well as composite microscopic particles such as atoms and molecules....
, in the quantum mechanical

Quantum mechanics

Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to 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

Gibbs paradox

Originally considered by Josiah Willard Gibbs in his paper On the Equilibrium of Heterogeneous Substances, the Gibbs paradox applies to thermodynamics....
.

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

Potts model

In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spin on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics....
.

Grand canonical partition function

Definition

In a manner similar to the definition of the canonical partition function for the canonical ensemble

Canonical ensemble

A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system....
, we can define a grand canonical partition function for a grand canonical ensemble

Grand canonical ensemble

In statistical mechanics, the grand canonical ensemble is a statistical ensemble , where each system is in equilibrium with an external reservoir with respect to both particle and energy exchange....
, a system that can exchange both heat and particles with the environment, which has a constant temperature T, and a chemical potential

Chemical potential

In thermodynamics, physics and chemistry, chemical potential, symbolized by ?, is a term introduced by the American engineer, chemist and mathematical physicist Willard Gibbs, which he defined as follows:...
µ. The grand canonical partition function, although conceptually more involved, simplifies the theoretical handling of quantum systems because it incorporates in a simple way the spin-statistics of the particles(i.e. whether particles are bosons or fermions). A canonical partition function, which has a given number of particles is in fact difficult to write down because of spin statistics.

The grand canonical partition function for an ideal quantum gas, i.e. a gas of non-interacting particles in a given potential well is given by the following expression:

where N is the total number of particles in the gas, index i runs over every microstate (i.e. a single particle state in the potential) with n_i being the number of particles occupying microstate i and e_i being the energy of a particle in that microstate. is the set of all possible occupation numbers for each of these microstates such that S_in_i = N.

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

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

Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
requires that the occupation numbers only be 0 or 1, again adding up to N.

Specific expressions

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

(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 boson

Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
s:

and for a system composed of fermion

Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
s:

For the case of a Maxwell-Boltzmann gas, we must use "correct Boltzmann counting" and divide the Boltzmann factor by n_i!.

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.

Defining a=-ßµ, the most probable occupation numbers are:

For Boltzmann particles this yields:

For bosons:

For fermions:

which are just the results found using the canonical ensemble for Maxwell-Boltzmann statistics, Bose-Einstein statistics and Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
, 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

Variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
in total number of particles

Internal energy

Variance
Variance

In probability theory and statistics, the variance of a random variable, probability distribution, or sample is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value ....
in internal energy
Internal energy

In thermodynamics, the internal energy of a thermodynamic system, or a physical body with well-defined dimension, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electricity energy of atoms within molecules or crysta...

Pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....

Mechanical equation of state
Equation of state

In physics and thermodynamics, an equation of state is a relation between thermodynamic variables. More specifically, an equation of state is a thermodynamic equations 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:

(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

Particle in a box

In physics, the particle in a box is a problem consisting of a single particle inside of an infinitely deep potential well, from which it cannot escape, and which loses no energy when it collides with the walls of the box....
model or particle in a harmonic oscillator

Harmonic oscillator

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 Hooke's law:...
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

Gas in a box

In quantum mechanics, 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 of molecules which do not interact with each other except for instantaneous thermalizing collisions....
and gas in a harmonic trap

Gas in a harmonic trap

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 that do not interact with each other except for instantaneous thermalizing collisions....
articles for a description of quantum fluids.)

These results may be used to construct the grand partition function to describe an ideal Bose gas

Bose gas

An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose-Einstein statistics....
or Fermi gas

Fermi gas

A Fermi gas, or Free electron gas, is a collection of non-interacting fermions. It is the quantum mechanics version of an ideal gas, for the case of fermionic particles....
, and can be used as well to describe a classical ideal gas

Ideal gas

The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
.

Partition function (statistical mechanics)

Canonical partition function

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

See also