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CRC Press eBooks

Concepts of Statistical Physics

P. F. Bortignon, A. Bracco, R.A. Broglia

Published Jun 14, 2019• 0 citations
Publisher

Abstract

A number of powerful methods exist in statistical mechanics to describe the interweaving of the variety of elementary modes of excitation taking place in a quantal many-body system. Among them, one can mention the Static Path approximation (cf., e.g., Feynmann and Hibbs ( 1965 )), the Monte-Carlo Shell Model Method (Koonin et al. ( 1997 )), and the thermal Green-function formalism (cf., e.g., Mahan ( 1981 )). The first two approaches have not yet been systematically used to describe nuclear motion at finite temperature. We shall briefly discuss them in the present chapter. The Matsubara techniques, which have been applied to the study of a large variety of phenomena, are discussed in Ch. 9 .

DOI
10.1201/9780203753224-9

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A number of powerful methods exist in statistical mechanics to describe the interweaving of the variety of elementary modes of excitation taking place in a quantal many-body system. Among them, one can mention the Static Path approximation (cf., e.g., Feynmann and Hibbs ( 1965 )), the Monte-Carlo Shell Model Method (Koonin et al. ( 1997 )), and the thermal Green-function formalism (cf., e.g., Mahan ( 1981 )). The first two approaches have not yet been systematically used to describe nuclear motion at finite temperature. We shall briefly discuss them in the present chapter. The Matsubara techniques, which have been applied to the study of a large variety of phenomena, are discussed in Ch. 9 .
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.
In this chapter, the authors consider the formal description of the response and of the correlations displayed by the nuclear many-particle system at finite temperature. They work out the theoretical tools which allow to describe mean field properties, of both single-particle and collective motion, as well as their interweaving in highly excited systems. The authors show that the analysis of the response of the system to a weak external probe, and the treatment of correlations proceed in much the same way as they did at zero temperature. There is, however, a main difference to be kept in mind. At finite temperature, one no longer finds a unique state which describes the system, and a statistical description is called for. To study a quantum mechanical system, it is generally convenient to use the grand canonical ensemble, for which the energy of the system and the number of particles are not required to have definite values but only particular average values.
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.
In this article we review a less known unperturbative and powerful many-body method in the framework of classical statistical mechanics and then we show how it works by means of explicit calculations for a nontrivial classical model. The formalism of two-time Green functions in classical statistical mechanics is presented in a form parallel to the well known quantum counterpart, focusing on the spectral properties which involve the important concept of spectral density. Furthermore, the general ingredients of the classical spectral density method (CSDM) are presented with insights for systematic nonperturbative approximations to study conveniently the macroscopic properties of a wide variety of classical many-body systems also involving phase transitions. The method is implemented by means of key ideas for exploring the spectrum of elementary excitations and the damping effects within a unified formalism. Then, the effectiveness of the CSDM is tested with explicit calculations for the classical ddd-dimensional spin-SSS Heisenberg ferromagnetic model with long-range exchange interactions decaying as r−pr^{-p}r−p (p>dp>dp>d) with distance rrr between spins and in the presence of an external magnetic field. The analysis of the thermodynamic and critical properties, performed by means of the CSDM to the lowest order of approximation, shows clearly that nontrivial results can be obtained in a relatively simple manner already to this lower stage. The basic spectral density equations for the next higher order level are also presented and the damping of elementary spin excitations in the low temperature regime is studied. The results appear in reasonable agreement with available exact ones and Monte Carlo simulations and this supports the CSDM as a promising method of investigation in classical many-body theory.