Table of Contents
- Preliminary Pages
- Preface
- Biographies
- List of Symbols
- 1. Introduction
- 2. Linear Irreversible Thermodynamics
- 3. The Microscopic Connection
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- 3.1 Classical Mechanics
- 3.2 Phase Space
- 3.3 Distribution Functions and the Liouville Equation
- 3.4 Ergodicity, Mixing and Lyapunov Exponents
- 3.5 Equilibrium Time Correlation Functions
- 3.6 Operator Identities
- 3.7 The Irving-Kirkwood Procedure
- 3.8 Instantaneous Microscopic Representation of Fluxes
- 3.9 The Kinetic Temperature
- References
- 4. The Green Kubo Relations
- 5. Linear Response Theory
- 6. Computer Simulation Algorithms
- 7. Nonlinear Response Theory
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- 7.1 Kubo’s Form for the Nonlinear Response
- 7.2 Kawasaki Distribution Function
- 7.3 The Transient Time Correlation Function Formalism
- 7.4 Trajectory Mappings
- 7.5. Numerical Results for the Transient Time-Correlation Function
- 7.6. Differential Response Functions
- 7.7 Numerical Results for the Kawasaki Representation
- 7.8 The Van Kampen Objection to Linear Response Theory
- References
- 8. Time Dependent Response Theory
-
- 8.1 Introduction
- 8.2 Time Evolution of Phase Variables
- 8.3 The Inverse Theorem
- 8.4 The Associative Law and Composition Theorem
- 8.5 Time Evolution of the Distribution Function
- 8.6 Time Ordered Exponentials
- 8.7 Schrödinger and Heisenberg Representations
- 8.8 The Dyson Equation
- 8.9 Relation Between p- and f- Propagators
- 8.10 Time Dependent Response Theory
- 8.11 Renormalisation
- 8.12 Discussion
- References
- 9. Steady State Fluctuations
- 10. Towards a Thermodynamics of Steady States
- Index
List of Figures
- Figure 1.1. Methods for determining the Shear viscosity
- Figure 2.1. The change in the mass contained in an arbitrary closed volume V can be calculated by integrating the mass flux through the enclosing surface S.
- Figure 2.2. Definition of the pressure tensor.
- Figure 2.3. Newton's Constitutive relation for shear flow.
- Figure 2.4. Frequency Dependent Viscosity of the Maxwell Model.
- Figure 2.5. The transient response of the Maxwell fluid to a step-function strain rate is the integral of the memory function for the model, η M(t).
- Figure 2.6. The transient response of the Maxwell model to a zero time delta function in the strain rate is the memory function itself, η M(t).
- Figure 3.1 Gauss' Principle of Least Constraint
- Figure 3.2 Phase Space Trajectory 6N-dimensional Γ-space. As time evolves the system traces out a trajectory in 6N-dimensional Γ-space.
- Figure 3.3 The Schrödinger-Heisenberg Equivalence
- Figure 3.4 Equilibrium time autocorrelation function of real variable A.
- Figure 3.5 The propagator is norm preserving
- Figure 4.1. The projection operator P, operating on B produces a vector which is the component of B parallel to A
- Figure 4.2. Schematic diagram of the frequency- and wavevector dependent viscosity and stress autocorrelation function. We can resolve the wavevector dependent momentum density into components which are parallel and orthogonal to the wavevector, k.
- Figure 4.3. The relationship between the viscosity, ῆ(k,ω), and the stress autocorrelation function,
. At k=0 both functions are identical. At ω=0 but k≠0, the stress autocorrelation function is identically zero. The
stress autocorrelation function is discontinuous at the origin. The viscosity is
continuous everywhere but non-analytic at the origin (see Evans, (1981)). - Figure 6.1. Orthogonal periodic boundary conditions
- Figure 6.2.
- Figure 6.3. Lees-Edwards periodic boundary conditions for planar Couette flow
- Figure 6.4. The sliding-brick and deforming-cube representations of Lees-Edwards boundary conditions are equivalent.
- Figure 6.5. A particle moving out of the top of a cell is replaced by its image from the cell below.
- Figure 6.6. SLLOD equations of motion give an exact representation of planar Couette flow.
- Figure 6.7. Viscosity of the N = 2048 WCA fluid.
- Figure 6.8. The internal energy of a fluid plotted as a function of the strain rate.
- Figure 6.9. Normal stress coefficients for the N = 2048 WCA fluid.
- Figure 6.10. Frequency-dependent shear viscosity at the Lennard-Jones triple point.
- Figure 6.11. High shear rate string phase in soft discs.
- Figure 6.12.Thermal conductivity: Lennard-Jones triple point.
- Figure 6.13. The colour conductivity as a function of the Laplace transform variable, s.
- Figure 6.14. A test of equation (6.121), for the Lennard-Jones triple-point fluid.
- Figure 6.15. The various Norton ensemble susceptibilities as a function of frequency. The system is the Lennard-Jones triple-point fluid.
- Figure 7.1 Reduced shear stress, P*xy=Pxy(σ2∕ε), in the two-dimensional soft-disc system. Pxy(T), calculated from the transient correlation function; Pxy(D), calculated directly.
- Figure 7.2 Direct (D) and transient (T) correlation function values of ∆p*=p*-p*0.
- Figure 7.3 The x-y element of the pressure tensor in the three-dimensional WCA system. T, TTCF prediction; D, direct simulation; GK, Green-Kubo prediction. SS, long-time steady-state stress computer using conventional NEMD.
- Figure 7.4 Shear dilatancy in three dimensions. For abbreviations see Fig 7.3
- Figure 7.5 Transient responses for the normal stress differences Pyy-Pzz and Pxx-Pyy for the three dimensional WCA system at a reduced strain rate of unity.
- Figure 7.6 We depict the systematic nonequilibrium response (the shaded curve) as the difference of the nonequilibrium response from the equilibrium response. By taking this difference we can dramatically reduce the noise in the computed systematic nonequilibrium response. To complete this calculation one averages this differenc over an ensemble of starting states.
- Figure 7.7 Illustration of the subtraction method.
- Figure 7.8 Shear stress for the three-dimensional WCA system at a strain rate of γ*=10-3. sub, subtraction technique; T, TTCF.
- Figure 7.9 Shear dilatancy for the three-dimensional WCA system at a strain rate of γ*=10-3. sub, subtraction technique; T, TTCF.
- Figure 7.10 We show computer simulation results for the Kawasaki normalization, Z(t). According to the Liouville equation this function should be unity for all times, t. This is clearly not the case (see Evans, 1990, for details).
- Figure 7.11 We compare four different methods of computing the nonlinear nonequilibrium response of a system of 18 soft discs to a suddenly imposed shear flow. The agreement between the Transient Time Correlation Function method and direct nonequilibrium molecular dynamics is better than 2 parts in 103 over the entire range of times studied. This is the most convincing numerical verification yet made of the correctness of the Transient Time Correlation Function method. The renormalized Kawasaki method (denoted Kawasaki) is in statistical agreement with the direct calculations but the bare Kawasaki method is clearly incorrect (see Evans, 1990, for details).
- Figure 7.12 Logarithm of the ensemble average of the phase space separation plotted as a function of reduced time for various values of the imposed shear rate, γ*
- Figure 7.13 We plot the ratio of the phase space separations as a function of strain rate and time. The ratios are computed relative to the separation at a reduced strain rate of 10-7. Curves denoted by 'av' are ensemble averages. Those not so denoted give the results for individual phase trajectories. Since the integrals of the Green-Kubo correlation functions converge to within a few percent by a reduced time of ~1.5, we see that the trajectory separation is varying linearly with respect to strain rate for reduced strain rates less than ~2. This is precisely the strain rate at which direct nonequilibrium molecular dynamics shows a departure of the computed shear viscosity from linear behaviour.
- Figure 8.1 We give a diagrammatic representation of the exchange of order of integrations in equation (8.8).
- Figure 10.1 The iterates of the quadratic map for some particular values of the parameter μ. The horizontal axis is xn and the vertical axis is xn+1. For μ=2 and 2.9 there is a single stable fixed point. For μ=3.3 there is a stable 2-cycle; for μ=3.5 a stable 4-cycle and for μ=3.561 a stable 8-cycle. The value μ=3.83 is in the period three window.
- Figure 10.2 The iterates of the quadratic map as a function of the parameter μ. The horizontal axis is the parameter 1≤μ≤4, and the vertical axis is the iterate 0≤xn≤1.
- Figure 10.3 The iterates of the quadratic map as a function of the parameter μ. This is an expanded version of Figure 10.2 to include more detail in the chaotic region. The horizontal axis is the parameter 3.5≤μ≤4, and the vertical axis is the iterate 0≤xn≤1. The windows of period three (at about μ=3.83), period five (at about μ=3.74), and period six (at about 3.63) are clearly visible.
- Figure 10.4 The distribution function for the iterates of the quadratic map in the chaotic region, at μ=3.65. The horizontal axis is the value of the iterate, and the vertical axis is the probability. Notice the distribution of narrow peaks which dominate the probability distribution.
- Figure 10.5 The distribution of iterates for the quadratic map at μ=4. The horizontal axis is the iterate, and the vertical axis is the probability. When correctly normalized, this agrees well will equation (10.15).
- Figure 10.6 The iterates of the Lorenz Model for a typical set of parameters which leads to chaotic behaviour. The iterates are the values obtained at the end of each 4th order Runge-Kutta step.
- Figure 10.7 We show the sum of the largest n exponents, plotted as a function of n, for three-dimensional 8-particle Couette flow at three different shear rates γ = 0, 1, and 2. The Kaplan-Yorke dimension is the n-axis intercept.
- Figure 10.8 The spectrum of phase space singularities for two dimensional 2 particle planar Couette flow at T*=1 and ρ*=0.4 as a function of γ. The function f(α) is the dimension of the set of points on the attractor that scale with exponent α. The range of singularities extends from 3 to αmin where the value of αmin decreases with increasing strain rate.
- Figure 10.9 The coordinate space distribution function for the relative position coordinate (x12,y12) at γ=1.25. The centre of the plot is the position of particle 1 (x1,y1), that is x12=y12=0. Notice that there is a preference for collisions to occur in the top right-hand side and lower left-hand side, and a significant depletion of counts near x12=0. ρ*=0.4,e*=0.25.
- Figure 10.10 The Lyapunov spectra for two-dimensional 4 particle planar Couette flow at T*=1 and ρ*=0.4. The open squares are for γ=0, the filled triangles are for γ=1 and the open circles are for γ=2. The Lyapunov spectra shifts downwards with increasing strain rate with the largest exponent shifting least. The sum of the exponents is zero at equilibrium and become more negative with increasing strain rate.
- Figure 10.11 The Lyapunov spectra for two-dimensional 2,4 and 8 particle equilibrium simulations at T*=1 and ρ*=0.4. The spectra are scaled so that the largest positive exponent occurs at the same exponent number regardless of system size. The open squares are for N=2, the filled circles for N=4 and the open circles for N=8.
- Figure 10.12 The Lyapunov spectra for two-dimensional 2,4 and 8 particle planar Couette flow at T*=1, ρ*=0.4, and γ=1.0. The spectra are scaled so that the largest positive exponent occurs at the same exponent number regardless of system size. The open squares are for N=2, the filled circles for N=4 and the open circles for N=8. The open squares are for N=2, the filled circles for N=4 and the open circles for N=8.
- Figure 10.13 The Lyapunov dimension for two-dimensional 2, 4 and 8-particle Couette flow at T*=1, ρ*=0.4, as a function of strain rate. The values of dimension are scaled with respect to the equilibrium dimension so that the y-axis represents the proportional change in dimension. The open squares are for N=2, the filled circles for N=4 and the open circles for N=8.
- Figure 10.14 Shows the pair distribution function for the 32-particle soft disc fluid at a relatively high reduced strain rate of 2.0. The reduced density and total energy per particle is 0.1, 1.921, respectively. The run length is 24 million timesteps. The distribution is, as far as can be told from the simulation data, completely smooth. In spite of the high anisotropy of this distribution, the configurational contribution to the system entropy is only about 0.4%.
- Figure 10.15 Shows the kinetic contribution to the system entropy as a function of strain rate. The system density is 0.1 and the energy per particle is 2.134. Within the accuracy of the data the entropy is essentially a linear function of strain rate. The derivative of the entropy with respect to strain rate gives ζ∕T. ζ is positive but decreases with strain rate, mostly due to the decrease in the thermodynamic temperature with increasing strain rate.
List of Tables
- Table 5.1 Linear Susceptibilities expressed as equilibrium time correlation functions‡
- Table 6.1 Green-Kubo Relations for Navier-Stokes transport coefficients.
- Table 6.2. Synthetic NEMD.
- Table 7.1 Mapping Parities
- Table 7.2 Summary of Phase Space Mappings
- Table 7.3. Strain rate dependent shear viscosities for the Triple Point Lennard-Jones fluid
- Table 7.4. Green Kubo equilibrium stress correlation function for shear viscosity
- Table 7.5. Exponential time Constants for phase separation in the Triple Point Lennard-Jones fluid under shear
- Table 10.1: Lyapunov exponents for two-body, two-dimensional Couette flow system.
- Table 10.2 Generalized dimensions for the two-body, two-dimensional Couette flow systems.
- Table 10.3. Equilibrium moderate density data‡
- Table 10.4. Low density data ‡
- Table 10.5. Nonequilibrium pressure: e=2.134, ρ=0.1