Über die Autorin bzw. den Autor

Luca Peliti is professor of statistical mechanics at the University of Naples Federico II in Italy. His books include "Biologically Inspired Physics".

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"This book provides a clear, no-nonsense approach to the basic ideas of the subject as well as an introduction to some of its modern applications. The main ideas are well illustrated by examples and exercises. There are important sections on numerical methods and dynamics, and a final chapter on complex systems gives the reader a foretaste of current research. The volume will serve as an excellent introductory graduate text for students in physics, chemistry, and biology."--John Cardy, University of Oxford

"Statistical mechanics has seen an extraordinary broadening of application in recent decades, from economics and the social sciences to computer science and biology.Statistical Mechanics in a Nutshell combines in one accessible book the main classical ideas of statistical mechanics with many recent developments. It should have a wide readership among young (and also less young) scientists seeking a clear view of modern statistical physics."--Bernard Derrida, école Normale Supérieure

"This superb text provides a balanced and thorough treatment of statistical physics. From thermodynamics and basic principles to renormalization group, dynamics, and complex systems, the presentation is a model of clarity, and the level of detail is highly appropriate for graduate students or advanced undergraduates. Each chapter concludes with a helpful list of recommended further reading. I see this becoming a standard textbook for the next generation of PhD students."--Daniel Arovas, University of California, San Diego

"This is an excellent and comprehensive introduction to statistical mechanics in all of its aspects. The exposition is stimulating and concise but always clear, avoiding pedantic details.Statistical Mechanics in a Nutshell has the potential to become a standard reference."--Giovanni Gallavotti, Sapienza University of Rome

Aus dem Klappentext

"This book provides a clear, no-nonsense approach to the basic ideas of the subject as well as an introduction to some of its modern applications. The main ideas are well illustrated by examples and exercises. There are important sections on numerical methods and dynamics, and a final chapter on complex systems gives the reader a foretaste of current research. The volume will serve as an excellent introductory graduate text for students in physics, chemistry, and biology."--John Cardy, University of Oxford

"Statistical mechanics has seen an extraordinary broadening of application in recent decades, from economics and the social sciences to computer science and biology.Statistical Mechanics in a Nutshell combines in one accessible book the main classical ideas of statistical mechanics with many recent developments. It should have a wide readership among young (and also less young) scientists seeking a clear view of modern statistical physics."--Bernard Derrida, école Normale Supérieure

"This superb text provides a balanced and thorough treatment of statistical physics. From thermodynamics and basic principles to renormalization group, dynamics, and complex systems, the presentation is a model of clarity, and the level of detail is highly appropriate for graduate students or advanced undergraduates. Each chapter concludes with a helpful list of recommended further reading. I see this becoming a standard textbook for the next generation of PhD students."--Daniel Arovas, University of California, San Diego

"This is an excellent and comprehensive introduction to statistical mechanics in all of its aspects. The exposition is stimulating and concise but always clear, avoiding pedantic details.Statistical Mechanics in a Nutshell has the potential to become a standard reference."--Giovanni Gallavotti, Sapienza University of Rome

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Statistical Mechanics in a Nutshell

PRINCETON UNIVERSITY PRESS

Contents

Preface to the English Edition................................xiPreface.......................................................xiii1 Introduction................................................12 Thermodynamics..............................................93 The Fundamental Postulate...................................554 Interaction-Free Systems....................................895 Phase Transitions...........................................1256 Renormalization Group.......................................1737 Classical Fluids............................................2158 Numerical Simulation........................................2519 Dynamics....................................................27710 Complex Systems............................................311Appendices....................................................357Appendix A Legendre Transformation............................359Appendix B Saddle Point Method................................364Appendix C A Probability Refresher............................369Appendix D Markov Chains......................................375Appendix E Fundamental Physical Constants.....................380Bibliography..................................................383Index.........................................................389

Chapter One

Lies, damned lies, and statistics. —Disraeli

1.1 The Subject Matter of Statistical Mechanics

The goal of statistical mechanics is to predict the macroscopic properties of bodies, most especially their thermodynamic properties, on the basis of their microscopic structure.

The macroscopic properties of greatest interest to statistical mechanics are those relating to thermodynamic equilibrium. As a consequence, the concept of thermodynamic equilibrium occupies a central position in the field. It is for this reason that we will first review some elements of thermodynamics, which will allow us to make the study of statistical mechanics clearer once we begin it. The examination of nonequilibrium states in statistical mechanics is a fairly recent development (except in the case of gases) and is currently the focus of intense research. We will omit it in this course, even though we will deal with properties that are time-dependent (but always related to thermodynamic equilibrium) in the chapter on dynamics.

The microscopic structure of systems examined by statistical mechanics can be described by means of mechanical models: for example, gases can be represented as systems of particles that interact by means of a phenomenologically determined potential. Other examples of mechanical models are those that represent polymers as a chain of interconnected particles, or the classical model of crystalline systems, in which particles are arranged in space according to a regular pattern, and oscillate around the minimum of the potential energy due to their mutual interaction. The models to be examined can be, and recently increasingly are, more abstract, however, and exhibit only a faint resemblance to the basic mechanical description (more specifically, to the quantum nature of matter). The explanation of the success of such abstract models is itself the topic of one of the more interesting chapters of statistical mechanics: the theory of universality and its foundation in the renormalization group.

The models of systems dealt with by statistical mechanics have some common characteristics. We are in any case dealing with systems with a large number of degrees of freedom: the reason lies in the corpuscular (atomic) nature of matter. Avogadro's constant, N_A = 6.02 10²³ mol^-1—in other words, the number of molecules contained in a gram- mole (mole)—provides us with an order of magnitude of the degrees of freedom contained in a thermodynamic system. The degrees of freedom that one considers should have more or less comparable effects on the global behavior of the system. This state of affairs excludes the application of the methods of statistical mechanics to cases in which a restricted number of degrees of freedom "dominates" the others—for example, in celestial mechanics, although the number of degrees of freedom of the planetary system is immense, an approximation in which each planet is considered as a particle is a good start. In this case, we can state that the translational degrees of freedom (three per planet)—possibly with the addition of the rotational degrees of freedom, also a finite number—dominate all others. These considerations also make attempts to apply statistical concepts to the human sciences problematic because, for instance, it is clear that, even if the behavior of a nation's political system includes a very high number of degrees of freedom, it is possible to identify some degrees of freedom that are disproportionately important compared to the rest. On the other hand, statistical methods can also be applied to systems that are not strictly speaking mechanical—for example, neural networks (understood as models of the brain's components), urban thoroughfares (traffic models), or problems of a geometric nature (percolation).

The simplest statistical mechanical model is that of a large number of identical particles, free of mutual interaction, inside a container with impenetrable and perfectly elastic walls. This is the model of the ideal gas, which describes the behavior of real gases quite well at low densities, and more specifically allows one to derive the well- known equation of state.

The introduction of pair interactions between the particles of the ideal gas allows us to obtain the standard model for simple fluids. Generally speaking, this model cannot be resolved exactly and is studied by means of perturbation or numerical techniques. It allows one to describe the behavior of real gases (especially noble gases), and the liquid-vapor transition (boiling and condensation).

The preceding models are of a classical (nonquantum) nature and can be applied only when the temperatures are not too low. The quantum effects that follow from the inability to distinguish particles are very important for phenomenology, and they can be dealt with at the introductory level if one omits interactions between particles. In this fashion, we obtain models for quantum gases, further divided into fermions or bosons, depending on the nature of the particles.

The model of noninteracting fermions describes the behavior of conduction electrons in metals fairly well (apart from the need to redefine certain parameters). Its thermodynamic properties are governed by the Pauli exclusion principle.

The model of noninteracting bosons has two important applications: radiating energy in a cavity (also known as black body) can be conceived as a set of particles (photons) that are bosonic in nature; moreover, helium (whose most common isotope, ⁴He, is bosonic in nature) exhibits, at low temperatures, a remarkable series of properties that can be interpreted on the basis of the noninteracting boson model. Actually, the transition of ⁴He to a superfluid state, also referred to as λ transition, is connected to the Einstein condensation, which occurs in a gas of noninteracting bosons at high densities. Obviously, interactions between helium atoms are not negligible, but their effect can be studied by means of analytic methods such as perturbation theory.

In many of the statistical models we will describe, however, the system's fundamental elements will not be "particles," and the fundamental degrees of freedom will not be mechanical (position and velocity or impulse). If we want to understand the origin of ferromagnetism, for example, we should isolate only those degrees of freedom that are relevant to the phenomenon being examined (the orientation of the electrons' magnetic moment) from all those that are otherwise pertinent to the material in question. Given this moment's quantum nature, it can assume a finite number of values. The simplest case is that in which there are only two values—in this fashion, we obtain a simple model of ferromagnetism, known as the Ising model, which is by far the most studied model in statistical mechanics. The ferromagnetic solid is therefore represented as a regular lattice in space, each point of which is associated with a degree of freedom, called spin, which can assume the values +1 and -1. This model allows one to describe the paramagnet— ferromagnet transition, as well as other similar transitions.

1.2 Statistical Postulates

The behavior of a mechanical system is determined not only by its structure, represented by motion equations, but also by its initial conditions. The laws of mechanics are therefore not sufficient by themselves to define the behavior of a mechanical system that contains a large number of degrees of freedom, in the absence of hypotheses about the relevant initial conditions. It is therefore necessary to complete the description of the system with some additional postulates—the statistical postulates in the strict sense of the word—that concern these initial conditions.

The path to arrive at the formulation of statistical postulates is fairly twisted. In the following section, we will discuss the relatively simple case of an ideal gas. We will formulate some statistical hypotheses on the distribution of the positions and velocities of particles in an ideal gas, following Maxwell's reasoning in a famous article [Maxw60], and we will see how from these hypotheses and the laws of mechanics it is possible to derive the equation of state of the ideal gas. What this argument does not prove is that the hypotheses made about the distribution of positions and velocities are compatible with the equations of motion—in other words, that if they are valid at a certain instant in time, they remain valid, following the system's natural evolution, also at every successive instant. One of Boltzmann's greatest contributions is to have asked this question in a clear way and to have made a bold attempt to answer it.

1.3 An Example: The Ideal Gas

1.3.1 Definition of the System

In the model of an ideal gas, one considers N point-like bodies (or particles, with mass equal to m, identical to one another), free of mutual interaction, inside a container of volume V whose walls are perfectly reflecting. The system's mechanical state is known when, for each particle, the position vector r and the velocity vector v are known. These vectors evolve according to the laws of mechanics.

1.3.2 Maxwell's Postulates

The assumption is that the vectors are distributed "randomly," and more specifically that:

1. The vectors pertaining to different particles are independent from one another. This hypothesis certainly does not apply, among other examples, to particles that are very close to each other, because the position of two particles that are very close is undoubtedly influenced by the forces that act between them. One can however expect that if the gas is very diluted, deviations from this hypothesis will have negligible consequences. If one accepts the hypothesis of independent particles, the system's state is determined when the number of particles dN whose position is located within a box, with sides dr = (dx, dy, dz) placed around a point defined by r = (x, y, z), and that are simultaneously driven by a velocity whose vector lies in a box defined by sides dv = (dv_x, dv_y, dv_z) around the vector v = (v_x, v_y, v_z): dN = f(r, v) dr dv is known. This defines the single-particle distribution f(r, v).

2. Position is independent of velocity (in the sense given by probability theory), and therefore the probability distribution f(r, v) is factorized: f(r, v) = f_r(r)f_v(v).

3. Density is uniform in the space occupied by the gas, and therefore fr(r) = N/V = t = const. if r is inside the container, and equal to zero otherwise.

4. The velocity components are mutually independent, and therefore f_v(v) = f_x(v_x)f_y (v_y)f_z(v_z).

5. The distribution f_v(v) is isotropic in velocity space, so that [f_v](v) will in actual fact depend only on the modulus v = | v | of v.

Exercise 1.1 Prove that the only distribution that satisfies postulates 4 and 5 is Gaussian:

f_v(v) ∞ exp (-λv², (1.1)

where λ is a constant, related to the average quadratic velocity of the particles:

<v²> = 3/2λ, (1.2)

and where therefore the average kinetic energy is given by

<1/2 mv²> = 3m/4λ. (1.3)

1.3.3 Equation of State

We will now prove that Maxwell's postulates allow us to derive the equation of state for ideal gases and provide a microscopic interpretation of absolute temperature in terms of kinetic energy.

Let us consider a particle of velocity v = (v_x, v_y, v_z) which, coming from the left, hits a wall, parallel to the plane (yz) (see figure 1.1). After the impact, it will be driven by velocity v' = (-v_x, v_y, v_z). The change Δp in its momentum p is given by Δp = p' - p = m(v' - v) = m(-2v_x, 0, 0). The number of impacts of this type that occur in a time interval Δt on a certain region of the wall of area S is equal to the number of particles driven by velocity v that are contained in a box of base equal to S and of height equal to v_x Δt. The volume of this box is equal to Sv_x Δt, and the number of these particles is equal to ρ f_v(v)v_x S Δt.

The total momentum ΔP transmitted from the wall to the gas, during the time interval Δt, is therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where i = (1, 0, 0) is the versor of axis x. In this expression, the integral over v_x runs only on the region v_x > 0 because only those particles that are moving toward the right contribute to pressure on the wall we are examining.

The total force that the wall exercises on the gas is given by F = -ΔP/Δt, and therefore the pressure p is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

In this equation, the factor 1/2 comes from the integration over v_x, which runs only on the region v_x > 0. It is well known that the equation of state for perfect gases takes the form

pV = nRT, (1.6)

where n = N/N_A is the number of moles, T is the absolute temperature, and R = 8.31 JK^-1 mol^-1 is the constant of the gas. By introducing the Boltzmann constant

k_b = R/N_A = 1.38 10^-23 JK^-1, (1.7)

and the particle density

ρ = N/V, (1.8)

it can be written as

p = ρ k_BT. (1.9)

If we compare this expression with equation (1.5), we obtain the constant λ:

λ = m/2k_BT. (1.10)

The Gaussian velocity distribution implies the following distribution of the magnitude v of the velocity, known as the Maxwell distribution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

where N is a normalization factor.

1.3.4 The Marcus and McFee Experiment

The Maxwell distribution can be measured directly by means of experiments on the molecular beams. We will follow the work by Marcus and McFee [Marc59]. A diagram of the experiment is given in figure 1.2. Potassium atoms are heated to a fairly high temperature (several hundred degrees Celsius) in an oven. The oven is equipped with a small opening from which the molecular beam departs. In the region traversed by the beam, a vacuum is maintained by a pump. Two rotating screens, set at a distance l from each other, act as velocity selectors. Each is endowed with a narrow gap, and they rotate in solidarity with angular velocity ω. The two gaps are out of phase by an angle φ. Therefore, only those particles driven by a velocity v = lω/φ will be able to pass through both gaps, hit the detector, and be counted. If we denote the total beam intensity by j₀, and the solid angle by which the detector is seen from the opening by dΩ, the number of particles driven by a velocity between v and v + dv that hit the detector in a given unit of time, is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

where N is a normalization factor.

(Continues...)

Excerpted from Statistical Mechanics in a Nutshellby Luca Peliti Copyright © 2003 by Bollati Boringhieri, Torino. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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