KINETIC THEORIES FOR STOCHASTIC MODELS OF LIQUIDS WITH HIGHLY COOPERATIVE DYNAMICS

Transcription

1 KINETIC THEORIES FOR STOCHASTIC MODELS OF LIQUIDS WITH HIGHLY COOPERATIVE DYNAMICS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF CHEMICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Steven Matthew Abel December 2008

2 c Copyright by Steven Matthew Abel 2009 All Rights Reserved ii

3 I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Hans C. Andersen) Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Curtis W. Frank) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Gerald G. Fuller) Approved for the University Committee on Graduate Studies. iii

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5 Abstract Liquids are often characterized by the highly correlated motion of constituent particles. Understanding the dynamics of particle motion can give deep insight into both microscopic and macroscopic properties of liquids, including the diffusive properties of labeled particles and macroscopic transport coefficients. In many dense fluids, correlated particle motion develops primarily as a result of caging phenomena, in which the motion of each particle is severely constrained by the presence of other nearby particles. Characterizing the dynamics of caging is of paramount importance in understanding the properties of many liquids, including atomic, polymeric, and colloidal fluids. In this thesis, to gain insight into caging phenomena in real liquids, we theoretically investigate two stochastic lattice models of liquids in which caging effects are important: a one-dimensional, single occupancy lattice gas and a lattice model of rigid rod polymers. Particles in both models undergo random motion on a lattice subject to the constraint that they cannot pass through one another. To characterize particle dynamics in these models, we develop a diagrammatic kinetic theory to study time correlation functions of density fluctuations. Correlation functions are represented as infinite sums of diagrams, and various approximations are constructed by evaluating subsets of the diagrams with certain physically or mathematically motivated features. The primary function of interest, the trace correlation function, is the time-dependent conditional probability distribution of the position of a labeled particle given its position at an earlier time. For each model, we derive diagrammatic series for the memory function and irreducible memory function. After showing that mean field approximations fail to provide an adequate description of the correlated particle dynamics, we develop approximations to the irreducible memory function motivated by the mode coupling theory of supercooled liquids [1]. Mode coupling theory is commonly regarded as a theory of caging, and testing the efficacy of mode coupling-like theories on models in which caging effects are important is a v

6 major goal of this work. Using both analytical and numerical methods, we solve the mode coupling-like equations self-consistently for the trace correlation function. Comparing the results with simulation data demonstrates that the theories are successful at describing, both qualitatively and quantitatively, many features of the models. A physical interpretation of the diagrams contributing to the theory provides additional insight into the dynamics of caging and may suggest how to develop theories of caging in more complicated and realistic models of liquids. vi

7 Acknowledgements The author is extremely grateful to Steve Tse, who provided the simulation results found in this thesis. Being able to compare theoretical predictions to simulation data greatly enhanced the theoretical development of this thesis. This work was supported by the National Science Foundation grant CHE vii

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9 Contents Abstract Acknowledgements v vii 1 Introduction Liquids and caging Correlation functions and kinetic theory Mode coupling theory Outline Stochastic dynamics and diagrammatics Stochastic dynamics and Hilbert space formalism Stochastic dynamics with detailed balance Dynamical variables Hilbert space formalism Density functions and the Boley basis State of the system The Boley condition Multipoint density functions Boley basis F n and K n functions Correlation functions Diagrammatic formalism Response function Structure and evaluation of diagrams ix

10 2.3.3 Memory functions and kinetic equations Lattice gas formulation Overview Properties of the lattice gas Equilibrium Dynamics The trace-dominant limit A comment on ergodicity Density variables Averages Equations of motion Diagrammatic series Vertices Trace correlation function Response function Memory functions and kinetic equations Memory function Irreducible memory function Scaled irreducible memory function Lattice gas results Quantities of interest Correlation functions Mean square displacement Fourier and Laplace transforms Kinetic equations Wavevector-time domain Wavevector-frequency domain Simple theories Short-time expansions Mean field theory Exact solution of C d Mode coupling theory x

11 4.5 Asymptotic solution of the kinetic equations Mobile dominant particles Immobile dominant particles Numerical solution of the kinetic equations Numerical method Results Other approximations Explicit approximation: free diffusion Explicit approximation: immobile particle Explicit approximation: subdiffusive particle Physical interpretation Rigid rods on a lattice Overview Properties of the model Equilibrium Semidilute regime Dynamics Trace-dominant limit Density variables Averages Equations of motion Diagrammatic series Vertices Trace correlation function Response function Memory functions and kinetic equations Memory function Irreducible memory function Scaled irreducible memory function Rigid rod results Scaling theory Kinetic equations xi

12 6.2.1 Trace correlation function Factorization of four-point functions Factorization of the correlation function Factorization Form of G irr xx Restriction of wavevector Numerical solutions of the kinetic equations Long rod approximation Limiting behavior Numerical solutions for C(k x, t) A Scaled memory function relations 99 A.1 Lattice gas A.1.1 Expansion of χ (1) M terms A.1.2 Compilation of terms A.2 Rigid rods A.3 Factorization of Q (c) B Asymptotic analysis 105 B.1 Mobile dominant particles B.2 Immobile dominant particles B.3 Explicit approximation to the mode coupling equations C Equations of motion and vertices 111 C.1 Equations of motion of density variables C.2 Vertices C.2.1 Q C.2.2 Q C.2.3 Q C.2.4 Q Bibliography 117 xii

13 List of Figures 1.1 In this schematic illustration of caging, the shaded particle is trapped by the presence of neighboring particles. In real liquids, motion of the caging particles is hindered by additional neighbors, and coordinated motion of many particles is necessary for large excursions by the trapped particle Diagrams in the series for χ 11. Labels are left off of the root points for clarity (i) Diagrams in the series for χ 11 with dashed boxes identifying the part of each diagram that contributes to the memory function. (ii) Diagrams contributing to the series for the memory function M(1, t; 1, t ) Examples of diagrams contributing to the memory function (S 1 = S 1 = S t). Dotted lines correspond to the trace line, solid lines correspond to unperturbed dominant propagators, and the vertex in the dashed box is a node Diagrams in the series for C t and C d. Root points on the right correspond to the origin at time zero. Root points on the left correspond to position R at time t Long-time mean square displacement with γ d = 0 as a function of concentration. Results include the exact analytic solution and the asymptotic solution to the mode coupling equations with η = 1 and η = MSD for c = 0.5 and α = γ d /γ t = 1. Solution to mode coupling equations with η = 2 (top figure) and η = 1 (bottom figure). Results include the numerical solution (solid line), asymptotic solution (dotted line), and simulation data (points, [17]) xiii

14 4.4 Trace correlation function for c = 0.5 and α = 1. Solution to mode coupling equations with η = 2 (top figure) and η = 1 (bottom figure). Results include the numerical solutions (solid line) and simulation data (points, [17]) for wavevectors (2π/200), 4(2π/200), 16(2π/200), and 64(2π/200) MSD for α = 1 at various concentrations. Numerical solution to mode coupling equations with η = 2 (solid lines). From top to bottom, values of concentration are c = 0.1, c = 0.5, and c = 0.9. Points correspond to simulation data [17] MSD for c = 0.5 and α = 10. Numerical (solid line) and asymptotic (dotted line) solutions to mode coupling equations with η = 2 (solid line), with simulation data (points, [17]) Trace correlation function for c = 0.5 and α = 10. Results include numerical solution to mode coupling equations with η = 2 and simulation data (points, [17]) for wavevectors (2π/800), 4(2π/800), 16(2π/800), 64(2π/800), and 256(2π/800) MSD for α = 0.1 at various concentrations. Numerical solution to mode coupling equations with η = 2 (solid lines), with asymptotic results (dotted line). From top to bottom, values of concentration are c = 0.1, c = 0.5, and c = 0.9. Points correspond to simulation data [17] MSD for c = 0.9 and α = Solution to mode coupling equations with η = 2 and η = 1. Results include the numerical solution (solid line), asymptotic solution (dotted line), and simulation data (points, [17]). The top line corresponds to the η = 1 case Trace correlation function for c = 0.9 and α = Solution to mode coupling equations with η = 2 (top figure) and η = 1 (bottom figure). Results include the numerical solutions (solid line) and simulation data (points, [17]) for wavevectors 4(2π/200), 16(2π/200), and 64(2π/200) MSD when dominant particles are immobile for various concentrations. Solution to mode coupling equations with η = 1 (solid lines) and η = 2 (dashed lines). From top to bottom, values of concentration are c = 0.1, c = 0.5, and c = 0.9. Points correspond to simulation data [17] xiv

15 4.12 Trace correlation function for c = 0.5 and α = 0. Solution to mode coupling equations with η = 2 (top figure) and η = 1 (bottom figure). Results include the numerical solutions (solid line) and simulation data (points, [17]) for wavevectors (2π/100), 4(2π/100), 16(2π/100), and 32(2π/100) Trace correlation function for c = 0.9 and α = 0. Solution to mode coupling equations with η = 2 (top figure) and η = 1 (bottom figure). Results include the numerical solutions (solid line) and simulation data (points, [17]) for wavevectors (2π/50), 4(2π/50), and 16(2π/50) MSD for trace particle with c = 0.5. Results include numerical (solid line) and asymptotic solution (dotted line) to explicit approximation with γ t = (1 c)γ t and η = 2, exact long-time solution (dashed line), and simulation data (points, [17]) MSD for trace particle with c = 0.5. Results include numerical solution to explicit approximation with γ t = 0 and η = 2 (dashed lines), numerical solution to mode coupling equations with η = 2 (solid lines), and simulation data (points, [17]) Trace correlation function for c = 0.5 at wavevectors (2π/200), 4(2π/200), 16(2π/200), and 64(2π/200). Results include numerical solution to explicit approximation with γ t = 0 and η = 2 (dashed lines), numerical solution to mode coupling equations with η = 2 (solid lines), and simulation data (points, [17]) MSD orthogonal to the trace rod, L = 300 and λ = cl 3 = 56. Results include numerical solution with η = 2 (solid line), simulation results (points, [17]), and free diffusion (dotted line) MSD orthogonal to the trace rod, L = 50 and λ = cl 3 = 15. Results include numerical solution with η = 2 (solid line), simulation results (points, [17]), and free diffusion (dotted line) MSD orthogonal to the trace rod, L = 50 and λ = cl 3 = 5. Results include numerical solution with η = 2 (solid line), simulation results (points, [17]), and free diffusion (dotted line) xv

16 6.4 MSD for (i) L = 300, λ = cl 3 = 56 (top figure) and (ii) L = 50, λ = 15 (bottom figure). Results include the numerical solution to the long rod approximation (solid lines), long-time limiting solution for η = 1 (dashed line), free diffusion (dotted line), and simulation data (points, [17]). Numerical results for η = 1 (top curve in each figure) and η = 2 (bottom curve) are shown xvi

17 Chapter 1 Introduction 1.1 Liquids and caging Liquids are often characterized by the highly correlated motion of constituent particles. Understanding the dynamics of particle motion can give deep insight into both microscopic and macroscopic properties of liquids, including the diffusive properties of labeled particles and macroscopic transport coefficients. In many dense liquids, short-ranged repulsive potentials between particles dominate the microscopic structure and dynamics. In these fluids, correlated particle motion develops primarily as a result of caging phenomena, in which the motion of each particle is severely constrained by the presence of other nearby particles. Each particle is said to be surrounded by a cage of neighboring particles, and to move an appreciable distance, a collective rearrangement of the caging particles must occur. Figure 1.1: In this schematic illustration of caging, the shaded particle is trapped by the presence of neighboring particles. In real liquids, motion of the caging particles is hindered by additional neighbors, and coordinated motion of many particles is necessary for large excursions by the trapped particle. 1

18 2 CHAPTER 1. INTRODUCTION Characterizing the breakdown of a cage is of paramount importance in understanding the properties of many liquids, including atomic, polymeric, and colloidal fluids. For example, caging is thought to dominate the dynamics of liquids near the glass transition, and it plays an important role in polymer physics, where entanglement effects at the microscopic level affect macroscopic properties such as viscosity and elasticity. In this thesis, to gain insight into caging phenomena in real liquids, we theoretically investigate two stochastic lattice models of liquids in which caging effects are important: a one-dimensional, single occupancy lattice gas and a lattice model of rigid rod polymers. Particles in both of these models interact only through simple excluded volume constraints that prohibit two particles from sharing a common lattice site at the same time. This interaction mimics the short-ranged steric repulsion that particles experience in real liquids and leads to caging phenomena in which the cooperative motion of neighbors can facilitate the motion of a confined particle. While the lattice models we study are simple, they capture important physical characteristics of real liquids: particles undergo random motion (on sufficiently long time scales) subject to the constraint that they cannot pass through one another. Because these characteristics are central to caging, the models can elucidate physical processes in real liquids. However, their simplicity allows us to focus exclusively on the physics of caging without worrying about other complicating effects, and theories constructed for the simple lattice models may shed insight into how to develop theories for more realistic models of liquids. 1.2 Correlation functions and kinetic theory The microscopic dynamics of liquids, from simple models to real fluids, are naturally studied from the standpoint of time correlation functions. These functions give a quantitative measure of the correlation between dynamical properties of a system at various times, and can thus provide a detailed description of microscopic dynamics. Time correlation functions can be studied by analytical theory, numerical simulation, and scattering experiments, and provide a common language with which to compare results obtained by the various methods. Time correlation functions can also be related to macroscopic transport properties, and thus provide a link between the microscopic dynamics of a system and its measurable macroscopic properties [2]. The theoretical study and calculation of time correlation functions falls under the domain

19 1.3. MODE COUPLING THEORY 3 of kinetic theory, the study of macroscopic collections of particles starting from a microscopic description of their interactions. Kinetic theory was pioneered by Boltzmann and Maxwell in the 19 th century, although contributions by Mori in the 1960s form the basis for much of modern kinetic theory. In particular, Mori introduced the concept of a Hilbert space of dynamical variables, projection operator techniques, and the language of memory functions that we use throughout this thesis [3, 4]. To understand the dynamics of caging in our model systems, we study time correlation functions using a formally exact diagrammatic kinetic theory developed by Andersen for classical fluids near equilibrium [5, 6, 7]. The theory expresses correlation functions and their related memory functions as infinite sums of diagrams, and has a formal structure similar to the Mayer cluster theory of equilibrium fluids. The theory is useful in a number of ways: unwieldy mathematical expressions are replaced by diagrams, and certain mathematical operations can be performed on the diagrams without reference to the underlying expressions; one can systematically develop approximations for correlation functions by summing physically or mathematically motivated subsets of diagrams; and diagrams can elucidate relationships between correlation functions and memory functions, suggesting ways to derive integral equations relating the two types of functions. The methods developed by Andersen for classical fluids can also be applied to models governed by stochastic dynamics. For example, Pitts and Andersen developed a diagrammatic formalism to study kinetic Ising models [8], and Feng and Andersen extended the results derived for classical fluids to a multidimensional lattice gas [9, 10]. 1.3 Mode coupling theory The purpose of using a diagrammatic theory in this thesis is to develop approximations for correlation functions that shed insight into caging. We explore various approximations, but focus particularly on those that assume the form of a mode coupling theory. The mode coupling theory of supercooled liquids was developed by Götze and co-workers in the early 1980s as a closure approximation for a hierarchy of memory function equations describing the behavior of a correlation function [1]. The approximation yields a selfconsistent integral equation for the correlation function, and successfully describes many aspects of liquid state dynamics near the glass transition. The theory predicts slow decay of the correlation function, and in some cases predicts an ergodic to nonergodic transition

20 4 CHAPTER 1. INTRODUCTION once thought to be a signature of the glass transition in real liquids. However, the theory is now thought to describe the onset of caging, not the glass transition itself. In spite of the large body of work investigating mode coupling theory, it is a mathematically motivated theory, and the underlying physical nature of the approximations is still unclear. The diagrammatic formulation of the kinetic equations facilitates the construction of mode coupling-like integral equations. We are interested in studying mode coupling approximations for the simple lattice models of interest since caging effects are important, and understanding physical features of the diagrams may give deeper insight into the physical nature of mode coupling theory. 1.4 Outline In Chapter 2, we discuss formal elements of the stochastic dynamics and diagrammatic theory that are common to both lattice models. We begin by considering a stochastic process on a discrete state space. A transition rate matrix governs the dynamics of the system and is used to construct a Liouville matrix that appears in the equation of motion of a dynamical variable. We then map the set of all dynamical variables onto a Hilbert space, and identify a set of multipoint density functions that are relevant to both models. We then construct the Boley basis, a set of orthogonal basis functions for the portion of the Hilbert space spanned by the multipoint densities. The time derivative of these functions are expressed in terms of Q-vertices, which are fundamental components in the diagrammatic theory. Identifying correlation functions of Boley basis functions as the correlation functions of interest, we construct a diagrammatic series for the response function that appears in the formal solution for the time correlation function. In Chapter 3, we introduce the one-dimensional lattice gas model in detail. After describing the physical content of the model, a transition rate matrix consistent with the dynamics is constructed. We use this to explicitly calculate the vertices that appear in our diagrammatic theory and obtain a formally exact diagrammatic series for the correlation functions of interest. From this we identify diagrams with specific topology, which leads to a discussion of memory functions and irreducible memory functions. We finally obtain a set of integro-differential equations relating the correlation function, memory function, and irreducible memory function.

21 1.4. OUTLINE 5 Chapter 4 discusses various approximations for the correlation function of a tagged particle. We first introduce simple approximations that do not correctly describe the behavior of the correlation function in the hydrodynamic regime and discuss the physics behind the results. We then turn to the memory function equations and introduce a simple mode coupling-like approximation in which parts of the irreducible memory function are replaced by the correlation function of a tagged particle. This gives a self-consistent set of memory function equations that we analyze both asymptotically and numerically. Results for the correlation function and mean square displacement of a tagged particle are compared against simulation data. Results from the mode coupling theory suggest a series of additional approximations that we then evaluate. In Chapters 5 and 6, we develop a similar theoretical framework to study the stochastic dynamics of rigid rod polymers on a cubic lattice. We derive a diagrammatic series for the correlation function of a tagged particle and obtain mode coupling-like equations that we then solve numerically. The results are again compared with simulation data, giving additional insight into the physics of caging and mode coupling theory.

22 6 CHAPTER 1. INTRODUCTION

23 Chapter 2 Stochastic dynamics and diagrammatic formalism In this thesis, we study two stochastic lattice models in which caging effects are important. This chapter discusses features of the stochastic dynamics and diagrammatic theory common to both. 2.1 Stochastic dynamics and Hilbert space formalism As discussed in the introduction, the use of Hilbert space methods has a rich history in classical statistical mechanics, beginning with Mori s pioneering work in the 1960s [3, 4]. The use of a Hilbert space formalism for stochastic processes with detailed balance was discussed in detail by Pitts and Andersen [8, 11]. This section reviews the features of the formalism relevant to this work Stochastic dynamics with detailed balance Consider a dynamical system with state space Ω, and let Γ Ω and Γ Ω represent distinct states of the system. In the simple lattice models of interest, the state space is discrete and finite in size, and a point in state space is specified by listing the location and other identifying features of each particle. The time evolution of the system is governed by a continuous time Markov chain in which W (Γ, Γ )dt is the conditional probability that the system is in state Γ after a differential 7

24 8 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS time interval dt given that the system was in state Γ at the beginning of the interval. The quantity W (Γ, Γ ) is called the transition rate from state Γ to state Γ, and W can be regarded as a matrix of transition rates between any two states of the system. We adopt the convention that the diagonal entries of the matrix are zero, W (Γ, Γ) = 0, noting that the probability of staying in state Γ after a differential time interval is 1 Γ W (Γ, Γ )dt To define a model system, it is necessary to specify both the state space Ω and the transition rates between all distinct elements of the state space. We are interested in studying systems at equilibrium, and thus restrict our attention to dynamics in which there exists a stationary distribution P (Γ) such that the detailed balance condition is satisfied: W (Γ, Γ )P (Γ ) = W (Γ, Γ)P (Γ) Γ, Γ Ω In the language of statistical mechanics, P (Γ) is the equilibrium distribution of the system, and the detailed balance condition guarantees that the distribution is stationary in time. While the transition rate matrix gives the instantaneous rate at which the system transitions from one state to another, we are ultimately interested in the time evolution of the system. To quantify this, let U(Γ, Γ ; t) denote the conditional probability that the system is in state Γ at time t given that the system was initially in state Γ. The rate of change of the conditional probability with respect to time is given by the master equation du(γ, Γ ; t) dt = ) (W (Γ, Γ )U(Γ, Γ ; t) W (Γ, Γ)U(Γ, Γ ; t) Γ = Γ L T (Γ, Γ )U(Γ, Γ ; t), where L T is the transpose of the matrix L(Γ, Γ ) = W (Γ, Γ) δ(γ, Γ ) Γ W (Γ, Γ) By analogy with classical statistical mechanics, we call this the Liouville matrix. It plays a central role in describing the equation of motion for a dynamical variable.

25 2.1. STOCHASTIC DYNAMICS AND HILBERT SPACE FORMALISM Dynamical variables A dynamical variable is a function of the state of the system. Let A(Γ) be a dynamical variable, and let A(Γ; t) denote the expected value of A at time t given the system was initially in state Γ. These functions can be related by the conditional probability, A(Γ; t) = Γ A(Γ )U(Γ, Γ; t) By taking the time derivative of A(Γ; t), one can obtain the following equation of motion for the dynamical variable in terms of the Liouville matrix da(γ; t) dt = Γ L(Γ, Γ )A(Γ ; t) The initial condition for this equation is A(Γ; 0) = A(Γ). A time correlation function of the two dynamical variables A and B is defined as the ensemble average 1 C AB (t) = B(Γ)A(Γ; t) = ( ) P (Γ) B(Γ)A(Γ; t) Γ = ( ) P (Γ) A(Γ )U(Γ, Γ; t)b(γ), Γ recalling that P (Γ) is the equilibrium distribution over the state space. Γ Hilbert space formalism The set of all dynamical variables of a dynamical system can be regarded as a vector space. Each function A(Γ) is associated with a unique vector in the vector space, A, and time dependent functions A(Γ; t) are mapped onto time dependent vectors A(t). The inner product of two vectors in the Hilbert space is given by the ensemble average of the product of the two functions that correspond to the vectors. For vectors A and B, the inner 1 In this thesis, all dynamical variables of interest are real-valued.

26 10 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS product is defined as A B = A(Γ)B(Γ) Functions of two state arguments are regarded as operators in the Hilbert space, V (Γ, Γ ) ˆV, where the operator is defined by its action on a vector in the Hilbert space. That is, given any vector A, the vector ˆV A corresponds to the function Γ V (Γ, Γ )A(Γ ) The Liouville operator, ˆL L(Γ, Γ ), is of central importance in understanding the time evolution of a system. Note that the equation of motion of a dynamical variable, previously written in terms of matrix multiplication, can be rewritten using the Hilbert space formalism as d A(t) dt = ˆL A(t) With the initial condition A(0) = A, the formal solution to this equation is A(t) = e tˆl A, where e tˆl Î + k=1 ( ) k tˆl k! Finally, using the Hilbert space formalism, the time correlation function C AB (t) can be rewritten C AB (t) = A(t) B = B e tˆl A

27 2.2. DENSITY FUNCTIONS AND THE BOLEY BASIS Density functions and the Boley basis The previous Hilbert space formalism can be applied to a wide array of stochastic models. In this section, we define the dynamical variables relevant to the study of particle dynamics in our simple models of caging. Points in single particle configuration space will have a different interpretation in the two models, but the definitions presented in this section hold for both State of the system In each model, a finite number of particles occupy a lattice and evolve in time according to stochastic dynamics satisfying the detailed balance condition. In the one-dimensional lattice gas, particles of various types each occupy one lattice site. In the rigid rod model, rods of various types and orientations occupy a fixed number of consecutive lattice sites. We generically refer to both lattice gas particles and rods as particles. The complete state of the system, Γ, is specified by listing the location and any other identifying features of each particle on the lattice Γ = { r 1, s 1 ; r 2, s 2 ; ; r N, s N } The total number of particles is N, r i r i (Γ) denotes the position of particle i in state Γ, and s i s i (Γ) is the species label for particle i. In the lattice gas, s i specifies the particle type, while in the rigid rod model, s i specifies both the rod type and the rod orientation. A different state of the system can be written Γ = { r 1, s 1 ; r 2, s 2 ; ; r N, s N } where r i r i(γ ) denotes the position of particle i in state Γ and s i s i(γ ) contains all other identifying information about particle i in state Γ The Boley condition The equilibrium distribution of each system is governed by a single occupancy potential: states in which no lattice site is occupied by more than one particle have zero energy, while states in which any lattice site is occupied by more than one particle have infinite energy and are not allowed. Every allowed state is equally likely in the equilibrium distribution,

28 12 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS and the stochastic dynamics are consistent with each particle executing a random walk on the lattice subject to the constraint that no two particles can occupy a common lattice site at the same time. Thus, at any given time, the state of the system must be allowed by the equilibrium distribution, and no two particles may ever occupy a common lattice site at the same time. A point in single particle configuration space is specified by a position R and species label S. To simplify notation, we typically refer to a point in single particle configuration space as R 1 S 1 1. The Boley condition is specified with respect to a set of arguments from single particle configuration space, { 1 ; 2 ; ; m } { R 1 S 1 ; R 2 S 2 ; ; R m S m } Assuming this set corresponds to a collection of m particles with the given position and species arguments, if no two particles from the collection occupy a common lattice site, then arguments {1; ; m} satisfy the Boley condition; otherwise they do not. Equivalently, the list of arguments is said to satisfy the Boley condition if and only if a collection of particles with the given position and species arguments is allowed by the equilibrium distribution. Arguments satisfying the Boley condition are called Boley arguments. The Boley condition is important when constructing multipoint density functions Multipoint density functions For a system in state Γ, the density of a point in single particle configuration space is given by ψ 1(RS; Γ) = N δ(r, r i ) δ(s, s i ), i=1 This value of this function is unity if there is a particle with species label S at position R; otherwise it is zero. Using the notation R 1 S 1 1, the single point density can be written ψ 1(1) = N δ(1, i) i=1

29 2.2. DENSITY FUNCTIONS AND THE BOLEY BASIS 13 where δ(1, i) δ(r 1, r i ) δ(s 1, s i ) The single point density can be used to construct additional density functions that depend on more than one configuration point. For example, N N ψ 1(1)ψ 1(2) = δ(1, i)δ(2, j) i=1 j=1 N N = δ(1, 2) δ(1, i) + δ(1, i)δ(2, j) i=1 i j=1 = δ(1, 2)ψ 1(1) + ψ 2(12) The function ψ 2 is a two point density in configuration space. In general, we define n-point densities in configuration space as N ψ n(1 n) = δ(1, i 1 )δ(2, i 2 ) δ(n, i n ) i 1 i 2 =i n=1 The multipoint densities defined in this section are dynamical variables and can be regarded as vectors in the Hilbert space of dynamical variables. For completeness, define ψ 0 = 1, which can also be regarded as a dynamical variable Boley basis The set of multipoint density functions is sufficiently large to completely characterize particle dynamics in liquids. However, in the development of a diagrammatic kinetic theory, it is useful to construct a new set of density functions from the set of multipoint densities. These functions form a basis for the subspace of the Hilbert space that includes all possible multipoint density functions. The new functions can be represented as linear combinations of ψ -functions, but have convenient orthogonality properties and are related to the fluctuation of ψ -functions away from their equilibrium averages. The new basis is known as the fluctuation basis or the Boley basis.

30 14 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS The functions in the Boley basis, denoted φ, can be constructed iteratively. Let φ 0 = ψ 0 = 1 This is the basis vector for constant functions, and S 0 is defined as the subspace spanned by φ 0. Subsequent basis vectors are constructed using a procedure similar to the Gram- Schmidt orthogonalization procedure. Let φ 1 (1) be the part of ψ 1 (1) orthogonal to φ 0. If we let P 0 represent the projection operator onto the S 0 subspace, then φ 1 (1) = ψ 1(1) P 0 ψ 1(1) A φ 1 -function is defined for each possible configuration point, and the S 1 subspace is defined to be the subspace spanned by the set of φ 1 (1) vectors for all values of the argument. If P 1 is the projection operator onto the S 1 subspace, then φ 2 (12) = ( 1 P 1 P 0 ) ψ 2 (12) This vector is defined only for arguments 1 and 2 that satisfy the Boley condition. general, φ n (1 n) is defined as In φ n (1 n) = ( 1 P n 1 P 1 P 0 ) ψ n (1 n), where P m represents the projection operator onto the S m subspace. This vector is defined only for arguments that satisfy the Boley condition. Projection operators onto the various subspaces can be constructed explicitly. For example, the projection operator onto the S 0 subspace is particularly simple, P 0 = 1 1 Using this, we can explicitly determine the form of φ 1 φ 1 (1) = ψ 1(1) 1 1 ψ 1(1) = ψ 1(1) 1 ψ 1(1) The φ 1 -function is an important physical quantity, as it represents the fluctuation of a single

31 2.2. DENSITY FUNCTIONS AND THE BOLEY BASIS 15 point density away from its equilibrium average φ 1 (1) = ψ 1(1) ψ 1(1) Correlation functions constructed using this function form the basis for many kinetic theories F n and K n functions Equilibrium averages of density functions play a central role in our theory. Note that an immediate consequence of the construction of the Boley basis is that φn (1 n) φ m (1 m ) = 0 n m This holds since vectors in Boley subspace S n are orthogonal to vectors in any other subspace S m. As a particular case, note that the equilibrium average of any non-constant Boley function is zero. This holds since φn (1 n) = φ 0 φ n (1 n) = 0 The only nontrivial inner products of Boley functions involve vectors from the same subspace. Functions of this form are defined as F n (1... n; 1... n ) = φ n (1... n) φ n (1... n ) In the construction of the theory, it is useful to define a function that acts as the matrix inverse of F n. This function, denoted K n, plays an important role in the construction of projection operators and is defined by 1...n K n (1... n; 1... n )F n (1... n ; 1... n ) = I n (1... n; 1... n ) P(1...n ) ( ) P δ(1; 1 ) δ(n; n ) where 1...n denotes a sum over all distinct unordered sets of configuration arguments that satisfy the Boley condition and P(1...n ) denotes a sum over all n! permutations of

32 16 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS arguments {1... n }. Note that F n and K n are defined only for left arguments and right arguments that each satisfy the Boley condition Correlation functions The simplest nontrivial correlation function that can be constructed with φ-functions is C 11 (11 ; t) = φ 1 (1; t) φ 1 (1 ) In general, correlation functions constructed with functions from the Boley basis have the form C nm (1 n, t; 1 m ; t ) = φ n (1 n; t) φ m (1 m ; t ) 2.3 Diagrammatic formalism The aim of our kinetic theory is to compute correlation functions that provide insight into caging in liquids. The primary focus will be on the correlation function C 11 (11 ; t), which quantifies the correlation between a particle of species S 1 originally at R 1 and a particle of species S 1 at R 1 after time t. In this section, we review how the time dependence of φ n (1 n; t) can be solved in terms of a response function χ, which can be expressed diagrammatically. This immediately leads to a diagrammatic expression for the correlation function. The formal basis for this theory was established by Andersen [5, 6, 7] and extended to stochastic models by Feng and Andersen [9]. The results presented here correspond to the time ordered series developed by Feng and Andersen Response function We have seen previously that the time derivative of a dynamical variable can be expressed in terms of the Liouville operator, which is defined in terms of the transition matrix. In the stochastic models of interest, the transition matrices ultimately lead to equations of motion for ψ n that are linear in ψ -functions. Since each Boley function is a linear combination of ψ -functions, it follows that the equation of motion for φ n can be written as a linear combination of φ-functions. Following the work of Andersen, the time derivative of a Boley

33 2.3. DIAGRAMMATIC FORMALISM 17 function is written d dt φ n(1 n; t) = m=1 1 m Q nm (1 n; 1 m )φ m (1 m ; t) where 1 m denotes a sum over all distinct unordered sets of configuration arguments that satisfy the Boley condition. The Q nm functions are independent of time, and can be expressed as Q nm (1 n; 1 m ) = 1 m K m (1 m ; 1 m ) φ m (1 m ) L φ n (1 n) These functions are called coupling coefficients since they relate the time derivative of φ n to the value of φ m for all possible values of m. In future sections, we refer to Q-functions as vertices due to the role they play in the construction of diagrams. Note that Q nm vertices are defined only when the left arguments (1 n) satisfy the Boley condition and the right arguments (1 m ) satisfy the Boley condition. Given the equation of motion for φ n above, our goal is to construct a solution for φ n (1 n; t) subject to the initial condition φ n (1 n; 0) = φ n (1 n). The formal solution can be expressed as φ n (1 n; t) = m=1 1 m χ nm (1 n, t; 1 m, 0)φ m (1 m ) where χ nm is a response function that describes the a response at time t to an perturbation at time zero. We commonly refer to χ nm as a propagator, and will express it as a formally exact diagrammatic series soon. First note that the correlation function C nm can be expressed C nm (1 n, t; 1 m ; 0) = φ n (1 n; t) φ m (1 m ) = χ nr (1 n, t; 1 r, 0) φ r (1 r ) φ m (1 m ) r=1 1 r The inner product in the last equality is nonzero only when r = m, and the correlation

34 18 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS Figure 2.1: Diagrams in the series for χ 11. Labels are left off of the root points for clarity. function can be written C nm (1 n, t; 1 m ; 0) = 1 m χ nm (1 n, t; 1 m, 0) F m (1 m ; 1 m ) An important consequence of using the fluctuation basis is that F m is the only equilibrium property that appears when evaluating C nm. The solution for χ nm can be expressed diagrammatically as the following χ nm (1 n, t; 1 m, t ) = the sum of all topologically distinct matrix diagrams with (i) n left roots labeled (1 n) at time t; (ii) m right roots labeled (1 m ) at time t ; (iii) Q (c) vertices; (iv) χ (0) bonds; and (v) free points such that (i) each root is attached to a χ (0) bond; and (ii) each free point is attached to a χ (0) bond and a Q (c) -vertex. A more detailed description of this diagrammatic series is given in the next section. Examples of diagrams in this series for χ 11 (1, t; 1, t ) are shown in Figure Structure and evaluation of diagrams Diagrams in the series for the response function describe the time evolution of density fluctuations. Each diagram is built from a small collection of basic objects (lines, open circles, and closed circles), and can be given a physical interpretation. Note that the response function χ nm has two time arguments, with the earlier time appearing on the right and the later time appearing on the left. Diagrams have a similar characteristic and are read from right to left, with horizontal ordering implying an ordering in time: an open circle appearing to the right of another open circle corresponds to an event that occurs at an earlier time.

35 2.3. DIAGRAMMATIC FORMALISM 19 Each diagram starts with one or more small open circles on the right and ends with one or more small open circles on the left. These small open circles, called roots, correspond to the arguments of the function being described. Between the roots are χ (0) bonds, Q (c) vertices, and free points. The χ (0) bonds are unperturbed propagators, given by χ (0) (1t; 1 t ) = Θ(t t )δ(1, 1 ), where the Heaviside function Θ(t t ) is nonzero when t > t. This bond represents a fluctuation at configuration point 1 that propagates forward in time, from t to t, without undergoing an explicit Q (c) interaction. Note that the δ(1, 1 ) term forces the configuration argument on the left to be the same as the initial one on the right in any diagram that contributes to the response function. The Q (c) vertices are parts of the Q-coupling coefficients that satisfy the cluster property, which is defined by the following statement: A function of two or more arguments is said to have the cluster property if the function is always zero if any subset of the arguments is separated from all of the other arguments by a distance that is many multiples of the correlation length in the system. When computing Q-vertices for the lattice gas, it is possible to identify a small set of vertices that can be used to represent all possible vertices. These terms satisfy the cluster property and are denoted by Q (c) nm, where n specifies the number of left arguments and m specifies the number of right arguments. (We will explicitly calculate these vertices for the two models of interest in future chapters.) Each Q (c) nm vertex is represented by a large open circle with n free points attached on the left and m free points attached on the right. Free points are represented by small closed circles and represent dummy variables that are summed over when evaluating a diagram. With the basic objects introduced, diagrams in the response function satisfy the following rules. The right of each χ (0) bond must be attached to either a right root or the left of a Q (c) vertex, and the left of each χ (0) bond must be attached to either a left root or the right of a Q (c) vertex. Each free point is attached to a χ (0) bond and a Q (c) vertex. Finally, in any vertical time slice involving only χ (0) propagators, the right arguments of the free propagators must satisfy the Boley condition. This is an important feature, as it serves to restrict free point sums to be sums involving only Boley variables. When evaluating a diagram with n vertices, the numerical value of the diagram is a product of (t t ) n /n! and the value of each Q (c) vertex evaluated with the appropriate

36 20 CHAPTER 2. STOCHASTIC DYNAMICS AND DIAGRAMMATICS arguments Memory functions and kinetic equations The primary correlation function of interest, C 11 (11 ; t), is related to the χ 11 response function by C 11 (11 ; t) = χ 11 (1, t; 1, 0)F 1 (1 ; 1 ) 1 The formally exact diagrammatic series for χ 11 is a special case of the result given for χ nm earlier. Each diagram in the series for χ 11 (1, t; 1, 0) contains a single root on the left (at time t) and a single root on the right (at time 0). Examples of diagrams in this series are given in Figure 2.1. Consider all diagrams in the series with at least one Q (c) vertex, and focus on the leftmost vertex in each diagram. The vertex must have exactly one χ (0) bond attached on the left (this bond is also attached to the left root). The vertex must also have either one or more χ (0) bonds attached on the right. If there is one bond on the right, then the vertex is a Q (c) 11 interaction. If there is more than one bond on the right, then the vertex is a Q(c) 1n interaction, with n > 1. If this is the case, since the right root is attached to a single bond, there must exist a time at which there is at most a single bond as one moves toward the right from the Q (c) 1n vertex. Identify the first region encountered (moving to the right) that contains a single χ (0) bond; the bond is attached to the right of a Q (c) m1 vertex, with m > 1. We identify the region from the left argument of the Q 1n to the right argument of the Q m1 as a memory function, M(1, t 1 ; 2, t 2 ). See Figure 2.2 for examples. The memory function has the property that the removal of any single χ (0) bond does not break the diagram into disconnected pieces. The diagrammatic series for the memory function is The χ (M) nm M(1, t 1 ; 2, t 2 ) = the sum of all topologically distinct matrix diagrams with (i) one left root labeled 1 at time t 1 ; (ii) one right root labeled 2 at time t 2 ; (iii) a Q (c) 1n vertex (with n > 1) attached to the left root; (iv) a Q(c) m1 vertex (with m > 1) attached to the right root; (v) a χ (M) nm function; and (vi) free points, such that each free point is attached to both a Q (c) vertex and the χ (M) nm response function response function can be expressed as a diagrammatic series consisting of a subset of the diagrams that appear in the series for the χ nm response function: χ (M) nm (1... n, t 1 ; 1... m, t 2 ) =