Thermodynamics of the glassy state

Thermodynamics

of the Glassy State

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Series in Condensed Matter Physics

Series Editor:

D R Vij

Department of Physics, Kurukshetra University, India

Other titles in the series include:

One and Two Dimensional Fluids: Properties of Smectic, Lamellar and

Columnar Liquid Crystals

A Jakli, A Saupe

Theory of Superconductivity: From Weak to Strong Coupling

A S Alexandrov

The Magnetocaloric Effect and its Applications

A M Tishin, Y I Spichkin

Field Theories in Condensed Matter Physics

Sumathi Rao

Nonlinear Dynamics and Chaos in Semiconductors

K Aoki

Permanent Magnetism

R Skomski, J M D Coey

Modern Magnetooptics and Magnetooptical Materials

A K Zvezdin, V A Kotov

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Series in Condensed Matter Physics

New York London

Luca Leuzzi

INFM - National Research Council (CNR)

Italy

Theo M. Nieuwenhuizen

University of Amsterdam

the Netherlands

Thermodynamics

of the Glassy State

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CRC Press

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Library of Congress Cataloging‑in‑Publication Data

Leuzzi, L. (Luca), 1972‑

Thermodynamics of the glassy state / L. Leuzzi, T.M. Niewenhuizen.

p. cm. ‑‑ (Series in condensed matter physics)

Includes bibliographical references and index.

ISBN 978‑0‑7503‑0997‑4 (hardback : alk. paper)

1. Spin glasses. 2. Glass. I. Nieuwenhuizen, Theo M. II. Title.

QC176.8.S68L48 2006

530.4’13‑‑dc22 2007025699

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Preface

This book fills a hole in the literature on glassy systems.

In the last fifteen years great progress has been made on our theoretical

understanding of structural glasses. This is also due to the use of many ideas

and concepts that have been derived in the framework of spin glasses (i.e.,

amorphous magnets) that are a different kind of amorphous system.

Spin glasses have experienced a very rapid development starting thirty years

ago; this was mainly due to the existence of solvable, but interesting models,

with infinite range forces, that display rather complex behavior. The need

for analyzing a solvable model in all its aspects has pushed theoreticians to

forge analytical tools that have been useful in many other fields, among them

structural glasses. Later, solvable models also for glassy systems were intro￾duced and studied in great detail.

The injection of these new ideas, that partially formalized old arguments,

led to a global rethinking of all the properties of the glassy state, starting

from basic thermodynamics properties. However, this new point of view is

only presented in original papers and in specialized monographs dedicated to

more specific aspects of the glassy states.

This book presents a comprehensive account of the modern theory of glasses

starting from the basic principles, i.e., thermodynamics, and from the exper￾imental analysis of some among the most important consequences of thermo￾dynamics (i.e., the Maxwell relations).

Immediately after the Introduction, the book underlines one of the most

crucial properties of glasses at low temperature: the existence of two temper￾atures in these off-equilibrium systems. Thermodynamics must be modified

in a deep and nontrivial way in order take care of this new and unexpected

phenomenon, that is at the basis of the modified fluctuation dissipation rela￾tions that are appropriate for glassy systems.

These concepts are carefully investigated using solvable (or nearly solvable)

models in which many different subtle properties can be studied in detail. In

this way one can see in an explicit and immediate manner the physical origin

of the aging phenomenon that is one of the hallmarks of glassy behavior.

At this point the reader is ready to tackle the approach based on the po￾tential energy landscape, whose features are discussed in general and studied

in simple models. Finally, more detailed theories of the glassy states are pre￾sented and analyzed where different microscopic mechanisms are discussed

also for realistic or quasi-realistic models of glasses.

V

VI

This book will certainly be extremely useful to anyone who approaches for

the first time the study of glasses because it first describes general properties

of the glassy states and later shows how these properties are present in specific

models: in this way the reader is not lost in a multitude of different models

that are used to derive general properties, as often happens in the literature.

This book will also be useful to the experienced researcher, who sometimes

may overlook the less technical consequences of his or her own work. It is

always very stimulating to read a well-done reflection on the basic results in a

developing field where the new conceptual points are discussed in a systematic

way. I am sure that this book will remain a reference text in the field for a

long time.

Giorgio Parisi

Rome, April 2007

Acknowledgements

The origin of this book goes back to 1996, when Giorgio Parisi, in a visit to

Amsterdam, drew attention to the paradox concerning the Prigogine-Defay

ratio in the traditional thermodynamic description of glass. We thank him for

constructive interactions and support throughout the years. Special thanks

are also due to Bernard de Jong, for encouragement and advice with the start

of the book.

In the course of our research on the subject and the writing of this book,

we have benefited from the interaction with many friends, colleagues and col￾laborators, of whom we mention, in alphabetical order, Armen Allahverdyan,

Luca Angelani, Gerardo Aquino, Andrea Baldassarri, Emanuela Bianchi, De￾sir´e Boll´e, Carlo Buontempo, Andrea Cavagna, Fabio Cecconi, Claudio Conti,

Andrea Crisanti, Leticia Cugliandolo, Silvio Franz, Adan Garriga, Alessan￾dra Gissi, Claude Godr`eche, Eric Hennes, John Hertz, Jorge Kurchan, Emilia

La Nave, Jean-Marc Luck, Luca Paretti, Maddalena Piazzo, Claudia Pombo,

Andrea Puglisi, Felix Ritort, Giancarlo Ruocco, David Saakian, Francesco

Sciortino, David Sherrington, Leendert Suttorp, Wim van Saarloos, Gerard

Wegdam, Emanuela Zaccarelli.

We also thank the students of the course “Theories and Phenomenology

of Structural Glass,” held at the Department of Physics of the University of

Rome “Sapienza,” in the academic years 2005-06, 2006-07 for their critical

observations and suggestions.

VII

VIII

IX

To Jurriaan,

Isabela

and Aurora

Acronyms

AG Adam-Gibbs

CRR Cooperative rearranging region

DB Disordered backgammon

FDR Fluctuation-dissipation ratio

FDT Fluctuation-dissipation theorem

FEL Free energy landscape

HNC Hyper-netted chain

HO Harmonic oscillators

HOSS Harmonic oscillators-spherical spins

IID Identically independently distributed

IS Inherent structure

LJ Lennard-Jones

LJBM Lennard-Jones binary mixture

LW Lewis-Wahnstr¨om

MC Monte Carlo

MCT Mode coupling theory

NM Narayanaswany-Moynihan

OTP Orthoterphenyl

PEL Potential energy landscape

PES Potential energy surface

PVC Polyvinylchloride

REM Random energy model

RFOT Random first order transition

RKKY Rudermann-Kittel-Kasuya-Yosida

ROM Random orthogonal model

SCE Small cage expansion

SPC/E Simple point charge extended

SSBM Soft spheres binary mixture

TAP Thouless-Anderson-Palmer

TTI Time translation invariant

VF Vogel-Fulcher

XI

Symbols

α: slow processes carrying the structural relaxation in the glass

αT : coefficient of thermal expansion

α: localization parameter in the density functional approach

to the random first order transition theory

α: logarithm of the total number of PEL basins

β: processes occurring on short timescales with respect to

glass relaxation times

β: inverse temperature in units of the Boltzmann constant kB

χ: susceptibility, integrated response

G: Gibbs free enthalpy

G: response function

G: shear modulus

K: fragility index

kB: Boltzmann constant

R: gas constant

Sc: configurational entropy

T0: Vogel-Fulcher temperature

Td: dynamic glass transition temperature, crossover temperature

Te: effective temperature

Tf : fictive temperature

Tf

: final temperature

(in Kovacs and PEL equilibrium matching protocols)

Tg: glass transition temperature, glass temperature

Tis: effective temperature in the framework of IS

TK: Kauzmann temperature

Tmc: mode-coupling temperature

τeq: relaxation time to equilibrium

τn: nucleation time

τobs, τexp: observation or experimental time

terg: time after which ergodicity is recovered

XIII