Statistical physics, mathematical problems in

2020 Mathematics Subject Classification: Primary: 82-01 [MSN][ZBL]

Problems that arise from the use of mathematical tools in statistical physics. Mathematical problems in statistical physics are basically related to two directions of the statistical theory: to equilibrium statistical mechanics, the problems of which are related to the development of methods for calculating averages using an equilibrium Gibbs distribution (see Statistical mechanics, mathematical problems in), and to non-equilibrium statistical physics, the difficulties of which lie in obtaining evolution equations for distribution functions that characterize the system at various stages of its development, and in solving them subsequently (see, for example, Kinetic equation; Brownian motion).

The problems of the mathematical methods of equilibrium statistical mechanics include the calculation of averages of the following types (when using a canonical Gibbs distribution):

$$ Z = \mathop{\rm Sp} e ^ {- H/ \theta } ,\ \ \langle A\rangle = \frac{1}{Z} \mathop{\rm Sp} Ae ^ {- H/ \theta } , $$

$$ \langle BA( t)\rangle = - \frac{1}{Z} \mathop{\rm Sp} BA( t) e ^ {- H/ \theta } , $$

etc., where $ H $ is the Hamiltonian of the system, $ \theta = kT $ is the temperature, $ A( t) $ is an operator in the Heisenberg representation, and $ Z $ is the partition function related to the free energy of the system by the relation $ F = - \theta \mathop{\rm ln} Z $( when a grand canonical distribution is used, instead of the operator $ H $, the operator $ H- \mu N $ figures in the process, where $ \mu $ is the chemical potential and $ N $ is the number of particles; instead of $ Z $ there is the grand partition function (cf. Statistical sum); instead of $ F $, the thermodynamic potential $ \Omega = F- \mu N $, etc.).

The calculation of the time-independent averages $ Z $ and $ \langle A\rangle $ solves the problems of the equilibrium theory (all equilibrium characteristics, such as internal energy, heat capacity, state equations, static response, etc., are defined by methods of thermodynamics based on the free energy $ F $), as well as of fluctuation theory; the calculation of quantities of the type $ \langle BA( t)\rangle $ enables one to study a whole series of dynamic (frequency-dependent) response functions of the system, transport coefficients, etc., as well as properties of the simplest perturbations of the system (generally when $ \theta \neq 0 $), their energy, damping, etc.

These averages can be calculated completely only in exceptional cases: for ideal systems and for certain special models. These calculations can serve as a zero approximation in further research. The most-frequently studied models of non-ideal statistical systems are systems with direct interaction between the particles (interaction of a finite radius, Coulomb interaction and others) or interaction between the particles and a photon-type field (in a solid body describing a thermal motion of the crystalline lattice), discrete Ising-type systems, systems of Heisenberg magnets with interaction between the nodes and with a finite action radius, and combinations of similar types of interactions. In a representation of second quantization, the Hamiltonian $ H = H _ {0} + H _ {1} $ is expressed through quadratic combinations of creation and annihilation operators (if the part $ H _ {0} $ is without interaction), through a four-form (if $ H _ {1} $ includes the direct interaction of particles), through a three-form of the type used in quantum electrodynamics (electron-photon interaction in $ H _ {1} $), etc.

Approximate methods for calculating these averages are based in most cases on the addition of corrections to the results obtained for the case $ H = H _ {0} $( if the zero approximation really is such a case in the physical sense), which take the form of a straightforward or modified expansion in powers of the parameter defining the strength of the interaction in the Hamiltonian $ H _ {1} $. When comparing the formal model under consideration with real systems, in many cases of practical interest the interaction parameter by which the "expansion" takes place proves to be not small. Difficulties of this type exist in the problem of electron gas in metals, in the theory of a non-ideal Bose gas, in the theory of fluids, in studying situations in the area of phase transitions or close to critical points, etc.

Leaving aside these problems of a formal nature, while bearing in mind that for certain particular cases a small parameter does exist, one sees that in developing the theory of perturbations with respect to this parameter, difficulties arise which are peculiar to many-body systems, and which become apparent when divergences emerge in the terms taking account of the many-particle correlations. Their emergence is caused by the fact that a simple series with integer powers of the "small" parameter being used, starting from a particular degree, does not reflect the real dependence of the required characteristic of the system on this parameter. These difficulties, being purely mathematical, are in principle surmountable. In order to detect the amendments, which "non-analytically" depend on the small parameter, methods have been worked out which are fundamentally related to the study of the classes of perturbations with respect to the interaction parameter and which are most essential for every concrete case of a formal series in perturbation theory.

In problems of statistical mechanics of classical systems with direct interaction between the particles, research in most cases is based on Bogolyubov's method [B] (see Bogolyubov chain of equations) or on Mayer's method [F]. In the first case, based on research of a chain of integro-differential equations for one-, two-, etc., particle distribution functions, the basic procedure for making an approximation is to cut off the chain. The higher-ranked distribution function in the integral part of the last equation in the chain is expressed using a combination of distribution functions of lower order. The procedure for such an expansion follows from an analysis of the physical singularities of the system and of the small parameter which is typical in the given situation. Thus, in systems of short-range forces of interaction, this parameter is the cube of the ratio of the radius of interaction to the mean distance between the particles (the foundation of the so-called virial expansion, or expansion by inverse degrees of the specific volume $ v = V/N $); in systems with Coulomb interaction, the parameter is related to the ratio of the mean energy of the Coulomb interaction of the particles to the mean of their kinetic energy, etc. Once the chain is cut off, the mathematical problem reduces to the solution of a system of equations which are non-linear in their integral part (or of one non-linear integral equation) relative to distribution functions subject to the normalization and weakening of correlations (the decomposition of many-particle functions, given a "separation" of their spatial arguments, into a product of functions of fewer particles), which play a part in particular boundary conditions.

The Mayer method (see [F]) for systems with short-range interaction potential

$$ \Phi _ {ij} = \Phi ( | \mathbf r _ {i} - \mathbf r _ {j} | ), $$

which has infinite repulsion over small distances, is based on the representation of the classical integral of states, $ Z $, in the form

$$ \frac{Z}{Z _ \mathop{\rm ID} } = \frac{1}{V ^ {N} } \int\limits _ { ( } V) d \mathbf r _ {1} \dots d \mathbf r _ {N} \prod _ { i\leq } j< j\leq N ( 1 + f _ {ij} ), $$

where the Mayer function $ f _ {ij} = \mathop{\rm exp} (- \Phi _ {ij} / \theta )- 1 $. For the specific free energy

$$ \frac{F}{N} = - \theta \mathop{\rm ln} Z ^ {1/N} $$

and for other characteristics of the system, virial expansions are obtained with coefficients expressed through integrals over products of an increasing number of functions $ f _ {ij} $, pairwise related by one of their two arguments $ \mathbf r _ {j} $ or $ \mathbf r _ {i} $( for real models of the potential $ \Phi _ {ij} $, these integrals are calculated using numerical methods). The mathematical problems which arise when a virial expansion is used are problems of working out methods for summing these series (even if only partially), so as to approach the domain of a phase transition from gas to liquid or a critical domain, as well as general problems of the convergence of the expansion itself.

In problems of statistical mechanics of non-ideal quantum systems, in which the operators of the dynamic variables are expressed in terms of quantum creation and annihilation operators, methods initially developed in quantum field theory have proved to be effective. The most common are zero-temperature many-time techniques and the method of finite-temperature two-time Green functions. In the first two approaches [HCB], the formal series of perturbation theory of the mutual interaction between the particles, or between the particles and any field, are in a way analogous to the corresponding expansions in quantum field theory (in finite-temperature techniques the role of "time" is played by an imaginary inverse temperature). Consequently, considerable progress has been made in these formalisms in the area of diagram representations of series of perturbation theory, which reduce to the study not of the initial particles, but of quasi-particles with renormalized energy and with finite damping which interact through a renormalized effective interaction (of vertex parts), etc. A system of integral equations for a Green function of quasi-particles and of vertex parts or effective interaction is made up in such a way, as a rule, that its solution would be equivalent to the calculation of a finite sequence of terms of the formal series of perturbation theory, chosen according to specific principles (for example, for a given concrete system of the strongest in each of the orders in perturbation theory).

In a finite-temperature two-time formalism, [B], [AGD]–[Z], similar problems arise when considering a Bogolyubov-type chain of linked equations for Green functions with an increasing number of "particles" , and when making up a closed system of integral equations (generally non-linear) resulting from the realization of the expansion procedure. Mathematical problems related to the study of these equations are, basically, restricted either by the search for specific asymptotics of their solutions, which are guaranteed in the approximate sense, or for the class of all diagrams, or for an expansion method.

These methods are used in studies on such systems as electron gases with Coulomb interaction (the calculation of only the most important contributions in each order according to perturbation theory is equivalent to the calculation of the screening of the initial interaction), low-density systems with short-range forces of interaction (the first stage in summing or an equivalent operation leads to the replacement of the initial interaction by an effective one, which is defined by solving an equation similar to the quantum mechanical equation for a $ t $- scattering matrix), an electron-photon system, a Heisenberg system of magnets, etc.

A number of problems of statistical mechanics permit an asymptotically-accurate examination (in the limiting statistical sense $ N \rightarrow \infty $, $ N/V = \textrm{ const } $); such are systems with factorized four-fold interaction, which generalize a model superconductive system of Bardeen–Cooper–Schrieffer type, and others. This technique (see ) is related to the construction of an approximating Hamiltonian which permits an exact solution, and to the subsequent proof of the asymptotic proximity of the results obtained by it to those which correspond to the initial system.

#### References[edit]

[B] | N.N. Bogolyubov, "Selected works" , 1 , Kiev (1969) (In Russian) MR1321817 MR1228446 MR1098017 MR1098016 MR0871873 MR0804234 MR0555732 MR0414297 Zbl 0226.01021

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[F] | G.W. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963) MR0151255 Zbl 0111.43802
[HCB] | J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, "Molecular theory of gases and liquids" , Wiley (1954) Zbl 0057.23402
[AGD] | A.A. Abrikosov, L.P. Gor'kov, I.E. Dzyaloshinskii, "Methods of quantum field theory in statistical physics" , Prentice-Hall (1963) (Translated from Russian) MR0157687 Zbl 0135.45003
[T] | S.V. Tyablikov, "Methods of the quantum theory of magnetism" , Plenum (1967) (Translated from Russian) MR0398393
[Z] | D.N. Zubarev, "Double-time Green functions in statistical physics" Soviet Physics Uspekhi , 3 (1960) pp. 320–345 Uspekhi Fiz. Nauk , 71 : 1 (1960) pp. 71–116 MR0122068
[Bjr] | N.N. Bogolyubov jr., "A method for studying model Hamiltonians" , Pergamon (1972) (Translated from Russian) MR0406286