Paper
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Nonequilibrium Dynamics in Low Dimensional Systems
Authors
M. R. Evans,R. A. Blythe
Sally-Ann Cooper,Claude Negrier,Rachid Zentar,dhahri Ahmed,Rashmi Chourasia,Laura Plant,Peter McDonald,Frederic Villieras,Dimitri Arvanitis,Ángel Ríos,Thomas Ihn,Evandro de Mello,Marion Gehlen,Carmen Marco De La Calle,Maria Ida De Michelis,Ailsa Hocking,James Wynn,Beth Middleton,Hans-Erik Claesson,Vladimir Sechovsky,Annick RUBBENS,Olaf Joehren,Zygmunt Lalak,JIH-HWA GUH,CHE-MING TENG,Charles H. V. Hoyle,Rafael Monsalve,Curtis Meyer,Robert Anson,Agustin Sanchez,Henri William NASSER,Viet Do,Nerea Zabala,Silvia Moretti,Laurent NOEL,Christopher Proud,Katharina Krenn,Robert Poulin,Gamal Esmat,Mark Blows,Antonino Romano,Adel BENALI,Deepika ,Piedad Virginia Fernández-Redondo,晓幸 王,Thomas Flower,Rolf K. Reed,N/A ,xianqiong zou,Laurie Ann Ximenez-Fyvie,Benjamin Weimann,Emmanuel Gritti,Stephen Warren,Kirsten Wolff,Gerry Polton,Hulya Dagdeviren,Kyungjun Choi,Carlos Santos-Burguete,Ian Nicholson,Jennifer Call Jones,Seimei Go,Khodadad Namiranian,hugo van heuverswyn,Rajat Thomas,Peter Day,rubens reis,Hung-Yi Chang,Stephen Whelan,Nathan Sanders,Antonio Pozzi,Shuo Zhang,Zelma Levi,Diane Hughes,Danny Cohn,Kerri Dorsey,Antonia Albrecht,Antonio Merla,Emiko Rimbara,André Pirralha,SADULLAH BAHAR,James Suliburk,Frank Coman,Raymond Townsend,Danilo Ribeiro,Yun Zhou,Dennis Hasselquist,Shinichi Fujimori,TSUGIYUKI MASUNAGA,Na Wang,Maria Soledad Diaz,Jennifer Moore,Martin Hermy,Syed Rizvi,Jason Rivers,Viktor Mikla,Huanliang Liu,Alessia Cedola,Peter Webb,Derek Smith,Jan Erik Henriksen,Robert Love,David Lunn,Laurence Rubenstein,William Sutherland,JENNIFER CHARLES,Hyo Hyun Ahn,Angeles Isabel Diaz Beltran,Reinhard Schumacher,Birgitta Heyman,Kiyoshi Tanida,Vivienne Wild,Xiang-Dong Li,William Clarkson,paula westerman,Alan Heavens,Miguel Camafort-Babkowski,Meghan Gray,Rosalinda Guevara-Guzman,David Lindsay,Itzell Gallardo-Ortíz,Bruce A Hungate,Cornelis de Haan,David Michayluk,Mark Maltman,Marek Gierlinski,Roberto Soria,David Keeling,Beth Katz,Valeriu Tudose,Robert White,Akio Hoshino,Elmar Veenendaal,Rainer Blasczyk,Guohua Cao,Alastair Edge,Gregory Daniel,Paige Bentley,Paolo Esposito,ELIZABETH WALMSLEY,Saara DeWalt,C de la Cuadra-Blanco,Zoe Kopp,Jon Wray,Guobao Zhang,Karri Koljonen,Henrik Pedersen,Ernest Williams,Prue Neath,Richard Balkin,Simon Platt,Clare Tait,Douglas Grindlay,cynthia stuhlmiller,Andrea Morandi,Paul O'Brien,Natacha Mesquita,Aleksay Golovchan,Sohair Sokkar,Alexandra D'Anglemont de Tassigny,YUSUF SARIOĞLU,Masahiro Fukuda,Neil Morton,Paul Jensen,Christian Michael Günther,Bridie Kent,Mingqun Lin,Stefanie Wagner,Elsa Cleland,Stuart Lumsden,Vitalii Turchenko,Vik Dhillon,George Delves,Susan KY Chow,Akihiro Matsumoto,Adam Scheinberg,john lawson,Jemai Dhahri,leticia vila,Victoria Parker,Miguel Humberto Soto Davila,Johannes Hirschberger,Danny Steeghs,Ingrid van Biezen,Philippe Jean Bousquet,Brett Ingram,Philippe Grandcolas,Stephen Glencross Brown,Kathryn Webert,Carlo Giocoli,Patrick Pollock,ANA TOMAS VIDAL,Gloria Thupayagale-Tshweneagae,Steven Presley,I-Wei Chen,Alessia Gualandris,Katy Bennett,Inga Zeisset,Shih-Lin Chang,X. 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arXiv:cond-mat/0110630v2 [cond-mat.stat-mech] 10 Dec 2001 Nonequilibrium Dynamics in Low
Dimensional Systems
M. R. Evans and R. A. Blythe 1
Department of Physics and Astronomy, University of Edinburgh, Mayfield Road,
Edinburgh EH9 3JZ, U.K.
Abstract
In these lectures we give an overview of nonequilibrium stochastic systems. In par-
ticular we discuss in detail two models, the asymmetric exclusion process and a
ballistic reaction model, that illustrate many general features of nonequilibrium
dynamics: for example coarsening dynamics and nonequilibrium phase transitions.
As a secondary theme we shall show how a common mathematical structure, the
q-deformed harmonic oscillator algebra, serves to furnish exact results for both sys-
tems. Thus the lectures also serve as a gentle introduction to things q-deformed.
Key words: Nonequilibrium Dynamics, Stochastic Processes, Phase Transition,
Asymmetric Exclusion Process, Reaction Kinetics
PACS: 02.50.-r, 05.40.-a, 05.70.Fh
1 Introduction
In these lectures we explore the subject of nonequilibrium dynamics. Before
getting into any kind of detail let us first establish what we mean by a nonequi-
librium system. This is best done by taking stock of our understanding of an
equilibrium system. Consider the Canonical (Boltzmann) distribution for a
systems with configurations labelled C each with energy E(C):
P (C) = exp(−βE(C))
Z (1)
Email addresses: m.evans@ed.ac.uk (M. R. Evans), r.a.blythe@ed.ac.uk (R.
A. Blythe).
1 Present address: Department of Physics and Astronomy, University of Manch-
ester, Manchester, M13 9PL, U.K.
Preprint submitted to Elsevier Science 1 November 2018
Dimensional Systems
M. R. Evans and R. A. Blythe 1
Department of Physics and Astronomy, University of Edinburgh, Mayfield Road,
Edinburgh EH9 3JZ, U.K.
Abstract
In these lectures we give an overview of nonequilibrium stochastic systems. In par-
ticular we discuss in detail two models, the asymmetric exclusion process and a
ballistic reaction model, that illustrate many general features of nonequilibrium
dynamics: for example coarsening dynamics and nonequilibrium phase transitions.
As a secondary theme we shall show how a common mathematical structure, the
q-deformed harmonic oscillator algebra, serves to furnish exact results for both sys-
tems. Thus the lectures also serve as a gentle introduction to things q-deformed.
Key words: Nonequilibrium Dynamics, Stochastic Processes, Phase Transition,
Asymmetric Exclusion Process, Reaction Kinetics
PACS: 02.50.-r, 05.40.-a, 05.70.Fh
1 Introduction
In these lectures we explore the subject of nonequilibrium dynamics. Before
getting into any kind of detail let us first establish what we mean by a nonequi-
librium system. This is best done by taking stock of our understanding of an
equilibrium system. Consider the Canonical (Boltzmann) distribution for a
systems with configurations labelled C each with energy E(C):
P (C) = exp(−βE(C))
Z (1)
Email addresses: m.evans@ed.ac.uk (M. R. Evans), r.a.blythe@ed.ac.uk (R.
A. Blythe).
1 Present address: Department of Physics and Astronomy, University of Manch-
ester, Manchester, M13 9PL, U.K.
Preprint submitted to Elsevier Science 1 November 2018
where β = 1/kT . The task is to calculate the partition function
Z = ∑
C
exp(−βE(C)) , (2)
from which all thermodynamic properties, in principle, can be computed. The
distribution (1) applies to systems in thermal equilibrium i.e. free to exchange
energy with an environment at temperature T . It can easily be generalised to
systems free to exchange particles, volume etc but it always relies on the
concept of the system being at equilibrium with its environment.
If one were interested in dynamics, for example to simulate the model on a
computer, one might choose transition rates between configurations to satisfy
W (C′ → C)e−βE(C′) = W (C → C′)e−βE(C) (3)
where W (C′ → C) is the transition rate from configuration C′ to C. Condi-
tion (3) is known as detailed balance and guarantees (under the assumption
of ergodicity to be discussed below) that starting from some nonequilibrium
initial condition the system will eventually reach the steady state of thermal
equilibrium given by (1). We will discuss further this dynamical relaxation
process and the properties of the steady state endowed by the detailed bal-
ance condition in Section 3. For the moment we note that a system relaxing
to thermal equilibrium is one realisation of a nonequilibrium system. In recent
years such relaxation dynamics have been of special interest, for example, in
the study of glassy dynamics whereby, on timescales realisable in experiment
(or simulation), the system never reaches the equilibrium state and it is a very
slowly evolving nonequilibrium state that is observed. This is sometimes re-
ferred to as ‘off-equilibrium’ dynamics. Also let us mention the field of domain
growth whereby an initially disordered state is quenched (reduced to a tem-
perature below the critical temperature for the ordered phase) and relaxes to
an ordered state through a process of coarsening of domains. The interesting
physics lies in the scaling regime of the coarsening process which is observed
before the equilibrium (ordered) state is reached.
The other meaning of nonequilibrium refers to a system that reaches a steady
state, but not a steady state of thermal equilibrium. Examples of such nonequi-
librium steady states are given by driven systems with open boundaries where
a mass current is driven through the system. Thus the system is driven by its
environment rather being in thermal equilibrium with its environment.
A pragmatic definition of a nonequilibrium system that encompasses all of
the scenarios above is as a model defined by its dynamics rather than any
energy function i.e. the configurations of the model are sampled through a local
stochastic dynamics which a priori does not have to obey detailed balance.
2
Z = ∑
C
exp(−βE(C)) , (2)
from which all thermodynamic properties, in principle, can be computed. The
distribution (1) applies to systems in thermal equilibrium i.e. free to exchange
energy with an environment at temperature T . It can easily be generalised to
systems free to exchange particles, volume etc but it always relies on the
concept of the system being at equilibrium with its environment.
If one were interested in dynamics, for example to simulate the model on a
computer, one might choose transition rates between configurations to satisfy
W (C′ → C)e−βE(C′) = W (C → C′)e−βE(C) (3)
where W (C′ → C) is the transition rate from configuration C′ to C. Condi-
tion (3) is known as detailed balance and guarantees (under the assumption
of ergodicity to be discussed below) that starting from some nonequilibrium
initial condition the system will eventually reach the steady state of thermal
equilibrium given by (1). We will discuss further this dynamical relaxation
process and the properties of the steady state endowed by the detailed bal-
ance condition in Section 3. For the moment we note that a system relaxing
to thermal equilibrium is one realisation of a nonequilibrium system. In recent
years such relaxation dynamics have been of special interest, for example, in
the study of glassy dynamics whereby, on timescales realisable in experiment
(or simulation), the system never reaches the equilibrium state and it is a very
slowly evolving nonequilibrium state that is observed. This is sometimes re-
ferred to as ‘off-equilibrium’ dynamics. Also let us mention the field of domain
growth whereby an initially disordered state is quenched (reduced to a tem-
perature below the critical temperature for the ordered phase) and relaxes to
an ordered state through a process of coarsening of domains. The interesting
physics lies in the scaling regime of the coarsening process which is observed
before the equilibrium (ordered) state is reached.
The other meaning of nonequilibrium refers to a system that reaches a steady
state, but not a steady state of thermal equilibrium. Examples of such nonequi-
librium steady states are given by driven systems with open boundaries where
a mass current is driven through the system. Thus the system is driven by its
environment rather being in thermal equilibrium with its environment.
A pragmatic definition of a nonequilibrium system that encompasses all of
the scenarios above is as a model defined by its dynamics rather than any
energy function i.e. the configurations of the model are sampled through a local
stochastic dynamics which a priori does not have to obey detailed balance.
2
1.1 Structure of these notes
These notes broadly follow the four lectures given at the summer school. In
addition a tutorial class was held to explore points left as exercises in the
lectures. In the present notes these exercises are included in a self-contained
form that should allow the reader to work through them without getting stuck
or else leave them for another time and continue with the main text. The notes
are structured as follows: in section 2 we give an overview of two simple models
that we are mainly concerned with in these lectures. In section 3 we then set
out the general theory of the type of stochastic model we are interested in and
point out the technical difficulties in calculating dynamical or even steady-
state properties. Section 4 is an interlude in which we introduce, in a self-
contained way, a mathematical tool—the q-deformed harmonic oscillator—
that will prove itself of use in the final two sections. In Section 5 we present
the solution of the partially asymmetric exclusion process and amongst other
things how the phase diagram (Figure 3) is generalised. In Section 6 we discuss
the exact solution of a stochastic ballistic annihilation and coalescence model.
2 Two simple models
In this work we will focus on two exemplars of nonequilibrium systems: the
partially asymmetric exclusion process and a particle reaction model. These
models have been well studied over the years and a large body of knowledge
has been built up [1]. We introduce the models at this point but will come
back to these models in more detail in Sections 5 and 6 in which we summarise
some recent analytical progress.
2.1 Asymmetric exclusion process
2.1.1 Model definition
The asymmetric simple exclusion process (ASEP) is a very simple driven lat-
tice gas with a hard core exclusion interaction [2]. Consider M particles on
a one-dimensional lattice of length N say. At each site of the lattice there is
either one particle or an empty site (to be referred to as a vacancy or hole)—
there is no multiple occupancy.
The dynamics are defined as follows: during each time interval ∆T each parti-
cle has probability ∆T of attempting a jump to its right and probability q∆T
of attempting a jump to its left; a jump can only succeed if the target site is
3
These notes broadly follow the four lectures given at the summer school. In
addition a tutorial class was held to explore points left as exercises in the
lectures. In the present notes these exercises are included in a self-contained
form that should allow the reader to work through them without getting stuck
or else leave them for another time and continue with the main text. The notes
are structured as follows: in section 2 we give an overview of two simple models
that we are mainly concerned with in these lectures. In section 3 we then set
out the general theory of the type of stochastic model we are interested in and
point out the technical difficulties in calculating dynamical or even steady-
state properties. Section 4 is an interlude in which we introduce, in a self-
contained way, a mathematical tool—the q-deformed harmonic oscillator—
that will prove itself of use in the final two sections. In Section 5 we present
the solution of the partially asymmetric exclusion process and amongst other
things how the phase diagram (Figure 3) is generalised. In Section 6 we discuss
the exact solution of a stochastic ballistic annihilation and coalescence model.
2 Two simple models
In this work we will focus on two exemplars of nonequilibrium systems: the
partially asymmetric exclusion process and a particle reaction model. These
models have been well studied over the years and a large body of knowledge
has been built up [1]. We introduce the models at this point but will come
back to these models in more detail in Sections 5 and 6 in which we summarise
some recent analytical progress.
2.1 Asymmetric exclusion process
2.1.1 Model definition
The asymmetric simple exclusion process (ASEP) is a very simple driven lat-
tice gas with a hard core exclusion interaction [2]. Consider M particles on
a one-dimensional lattice of length N say. At each site of the lattice there is
either one particle or an empty site (to be referred to as a vacancy or hole)—
there is no multiple occupancy.
The dynamics are defined as follows: during each time interval ∆T each parti-
cle has probability ∆T of attempting a jump to its right and probability q∆T
of attempting a jump to its left; a jump can only succeed if the target site is
3
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