Introduction and a Brief Review of Experimental Systems

Y. Imry

Published Feb 13, 1997• 0 citations

Abstract

Abstract 1. GENERALITIES Much of solid state theory and statistical physics is concerned with the properties of macroscopic systems. These are often considered while using the “thermodynamic limit” (system’s volume, Ω, and particle number, N, tending to infinity with n = N /Ω kept constant), which is a convenient mathematical device for obtaining bulk properties. Usually, the system approaches the macroscopic limit once its size is much larger than some correlation length ζ (or, more generally, than all such relevant lengths).

DOI: 10.1093/oso/9780195101676.003.0001

Related papers

Fundamentals of Algebraic Quantum Theory

Alfred Rieckers

Many body physics is concerned with macroscopic systems consisting of a very large number of microscopic particles (N ≈ 10 20 ). In this case the direct dynamical treatment of the complete system is impossible. A quite useful and frequently applied approximation consists in the neglection of terms which are relatively small as the volume of the systems tends to infinity, the particle density remaining constant. For the concise theoretical description of the typical many body phenomena such as phase transitions and spontaneous symmetry breakdown the use of the thermodynamical limit is even necessary, since no existing theory for the finite systems shows these effects. As a general rule one can say that a quantum system behaves like a thermodynamical system only in the infinite-volume limit. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vieweg+Teubner eBooksJan 1, 19802 citations

Statistical Mechanics and Critical Phenomena: A Brief Overview

S. B. Santra, Purusattam Ray

The basic concepts of statistical mechanics and its application to many-body systems, phase transitions and critical phenomena will be discussed briefly in this chapter. We will be using these ideas frequently in the subsequent lectures. Statistical mechanics describes the thermodynamic behaviour of macroscopic systems from the laws which govern the behaviour of the constituent elements at the microscopic level. The microscopic elements can be atoms, molecules, dipole moments or magnetic moments, etc. A macroscopic system is generally composed of a large number of these elements (of the order of Avogadro number N A ≈ 6.022 × 1023 per mole). Each element may have a large number of internal degrees of freedom associated with different types of motion such as translation, rotation, vibration etc. The constituent elements may interact with the external field applied to the system. There can also be very complex interaction among the constituent elements. The macroscopic properties of a system in the thermodynamic limit is thus determined by the properties of the constituent molecules, their interaction with external field as well as interaction among themselves. For a system of N elements, volume V and density ρ, the thermodynamic limit is defined as lim N → ∞, lim V → ∞ keeping ρ = N/V finite. In this limit, the extensive properties of the system become directly proportional to the size of the system (N or V), while the intensive properties become independent of the size of the system.

Texts and readings in physical sciencesJan 1, 20112 citations

Quantum statistical mechanics

David J. Toms

Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.

Cambridge University Press eBooksNov 15, 20073 citations

Introduction

J. W. Halleý

The problems of statistical mechanics are those which involve systems with a larger number of degrees of freedom than we can conveniently follow explicitly in experiment, theory or simulation. The number of degrees of freedom which can be followed explicitly in simulations has been changing very rapidly as computers and algorithms improve. However, it is important to note that, even if computers continue to improve at their present rate, characterized by Moore's “law,” scientists will not be able to use them for a very long time to predict many properties of nature by direct simulation of the fundamental microscopic laws of physics. This point is important enough to emphasize. Suppose that, T years from the present, a calculation requiring computation time t 0 at present will require computation time t ( T ) = t 0 2 − T /2 (Moore's “law,” see Figure 1). Currently, state of the art numerical solutions of the Schrödinger equation for a few hundred atoms can be carried out fast enough so that the motion of these atoms can be followed long enough to obtain thermodynamic properties. This is adequate if one wishes to predict properties of simple homogeneous gases, liquids or solids from first principles (as we will be discussing later). However, for many problems of current interest, one is interested in entities in which many more atoms need to be studied in order to obtain predictions of properties at the macroscopic level of a centimeter or more.

Cambridge University Press eBooksNov 16, 20060 citations

On Measurement

Bernard d’Espagnat

Macroscopic systems being composed of atoms and fields, there is no reason why they should not obey the quantum laws. It therefore seems quite natural to consider them as quantum systems, albeit extremely complex ones. Measurement constitutes a riddle, and a great many theories were put forward as attempts to solve this riddle. The number is so considerable indeed that to try and review them all would be an almost impossible undertaking. A few essentially quantum notions, such as quantum tunneling, have been shown quite indispensable for explaining a number of experimental facts concerning some systems of macroscopic size and complexity. Small differences necessarily exist between a system as it really is and its description by means of simple statements Moreover, elaborate experimental setups can in general be conceived of that, if used as measurement devices, would enhance these small differences.

CRC Press eBooksMay 4, 20180 citations

Introduction

Giuseppe Mussardo

Abstract In this chapter we introduce some general concepts of statistical mechanics and phase transitions, in order to give a rapid overview of the different topics of the subject and their physical relevance. For the sake of clarity and simplicity, we will focus our attention on magnetic systems but it should be stressed that the concepts discussed here are of a more general nature and can be applied to other systems as well. We will analyze, in particular, the significant role played by the correlation length in the phase transitions and the important properties of universality observed in those phenomena. As we will see, near a phase transition the thermodynamic quantities of a system present an anomalous power law behavior, parameterized by a set of critical exponents.

Aug 27, 20090 citations