Order, Disorder and Criticality

Advanced Problems of Phase Transition TheoryVolume 5

Yurij Holovatch

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813232112
• Topic: Physical Sciences
• Subtopic: Condensed Matter

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This book is the fifth volume of papers on advanced problems of phase transitions and critical phenomena, the first four volumes appeared in 2004, 2007, 2012, and 2015. It aims to compile reviews in those aspects of criticality and related subjects that are of current interest. The seven chapters discuss criticality of complex systems, where the new, emergent properties appear via collective behaviour of simple elements. Since all complex systems involve cooperative behaviour between many interconnected components, the field of phase transitions and critical phenomena provides a very natural conceptual and methodological framework for their study.

As the first four volumes, this book is based on the review lectures that were given in Lviv (Ukraine) at the "Ising lectures" — a traditional annual workshop on phase transitions and critical phenomena which aims to bring together scientists working in the field of phase transitions with university students and those who are interested in the subject.

--> Contents:

• Statistical Properties of One-Dimensional Directed Polymers in a Random Potential (V Dotsenko)
• Non-Euclidean Geometry in Nature (S Nechaev)
• Dynamics of Polymers: Classic Results and Recent Developments (M V Tamm and K Polovnikov)
• Generalized Ensemble Computer Simulations of Macromolecules (W Janke)
• Photo-Controllable Networks in Macromolecular Solutions and Blends (J M Ilnytskyi)
• Monte Carlo Methods for Massively Parallel Computers (M Weigel)
• Complex Networks and Infrastructural Grids (A Scala)

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--> Readership: Advanced undergraduates and graduate students, researchers and scientists interested in phase transitions and critical phenomena. -->
Keywords:Phase Transitions;Criticality;Scaling;Complex SystemsReview:0

Chapter 1

Statistical Properties of One-Dimensional Directed Polymers in a Random Potential

Victor Dotsenko^∗

LPTMC, Université Paris VI, 75252 Paris, France

This review is devoted to detailed consideration of the universal statistical properties of one-dimensional directed polymers in a random potential. In terms of the replica Bethe ansatz technique we derive several exact results for different types of the free energy probability distribution functions. In the second part of the review we discuss the problems which are still waiting for solutions.

Contents

1.Introduction

2.Replica method

3.Mapping to quantum bosons

4.Exact results for the free energy distribution functions

4.1.GUE Tracy-Widom distribution in the model with fixed boundary conditions

4.2.Inverse Laplace transformation approach

4.3.GOE Tracy-Widom distribution in the model with free boundary conditions

5.Zero-temperature limit

6.Random force Burgers turbulence

6.1.The model

6.2.Two-time velocity distribution function

6.3.Conclusions

7.Joint distribution function of free energies at two different temperatures

8.Conclusions

Appendix 1. GUE Tracy-Widom distribution function

Appendix 2. The Airy function integral relations

Appendix 3. Fredholm determinant with the Airy kernel

References

1.Introduction

It is well known that macroscopic characteristics of any random system defined in terms of macroscopic number of independent random parameters (according to the central limit theorem) are described by the Gaussian distribution function. In a sense, such type of the universal behavior is trivial, and not so much interesting. On the other hand, any non-trivial system usually requires individual consideration: although there are common universal macroscopic properties among microscopically different systems (e.g. scaling and critical phenomena at the phase transitions) until very recently no one would expect to have a universal function (different from the Gaussian one) which would describe macroscopic statistical properties of a whole class of non-trivial random systems.

One of the main achievement of the last three decades in the scope of the disordered systems is the discovery of the entire class of random systems whose macroscopic properties are described by the same universal probability distribution function which is called the Tracy-Widom (TW) distribution.¹ Traditionally it is called the KPZ universality class by the name of the celebrated Kardar-Parisi-Zang equation² which describes the time evolution of an interface (separating two homogeneous bulk phases) in the disordered inhomogeneous media.

Originally the solution of Tracy and Widom¹ was devoted to rather specific mathematical problem, namely the distribution function of the largest eigenvalue of N × N Hermitian matrices (Gaussian Unitary Ensemble (GUE)) in the limit N → ∞. Nowadays we have got rather comprehensive list of various systems (both purely mathematical and physical) whose macroscopic statistical properties are described by the same universal TW distribution function. Among these systems are: the longest increasing subsequences (LIS) model,³ zero-temperature lattice directed polymers with geometric disorder,⁴ the polynuclear growth (PNG) system,⁵ the oriented ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Contents
• 1. Statistical Properties of One-Dimensional Directed Polymers in a Random Potential
• 2. Non-Euclidean Geometry in Nature
• 3. Dynamics of Polymers: Classic Results and Recent Developments
• 4. Generalized Ensemble Computer Simulations of Macromolecules
• 5. Photo-Controllable Networks in Macromolecular Solutions and Blends
• 6. Monte Carlo Methods for Massively Parallel Computers
• 7. Complex Networks and Infrastructural Grids
• Subject Index