Theory of nonequilibrium transport based on a class of chaotic fluctuations

THEORY OF NONEQUILIBRIUM TRANSPORT BASED ON A CLASS OF CHAOTIC FLUCTUATIONS CHEW LOCK YUE A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2004 Acknowledgment First and foremost, I would like to express my heartfelt appreciation to my two thesis supervisors, Dr Christopher Ting and Prof Lai Choy Heng, for their supervision, patience, constant encouragement and invaluable advice I am most indebted to Chris for his guidance, and for being such a wonderful mentor He has ingrained in me the spirit of scientific research and has made my PHD journey a most enriching experience I am very grateful to Prof Lai for opening up the gateway to this journey and introducing me to the wonderful subject of chaos It is remarkable that despite his very busy schedule, his help is always forthcoming It is my pleasure to thank the organizers of the International Workshop and Seminar on Microscopic Chaos and Transport in Many-Particle Systems, and the Max-Planck-Institute for the Physics of Complex Systems, for their hospitality during my stay in Dresden It was there that I engaged in lively discussions on various aspects of chaos theory with Dr G Morriss and Dr T Taniguchi of the University of New South Wales, M Machida of the University of Tokyo, and Dr S Tasaki of Waseda University I had also benefited from enlightening discussions on the topic of Brownian ratchet with Dr R Klages of the Max-Planck-Institute for the Physics of Complex Systems and Dr P Hănggi of the University of Augsburg I am grateful to Dr H Kantz a of the Max-Planck-Institute for the Physics of Complex Systems for sharing with me his research papers relating to the transition of deterministic chaos to stochastic process Special thanks also go to J Davidson of the University of Toronto and M Bennett of the Georgia Institute of Technology for their stimulating discussion on the Kramers problem during the Dynamics Days Conference in Baltimore, Maryland Their friendship had made my stay in the cold winter of Maryland a warm one I i would also like to extend my gratitude to Dr C Beck of the University of London for his constructive comments and suggestions with regards to my research on dynamical Brownian motion and Gaussian diffusion processes, and to Dr Lou Jiann-Hua of the Department of Mathematics for giving me a rigorous mathematical perspective on the same area by being my teacher in stochastic calculus Deep appreciation goes to DSO National Laboratories for financial support and to Dr T Srokowski of the Institute of Nuclear Physics for his advice on some of the references Last but not least, I would like to express my greatest thanks to my beloved wife, Chui Mui for her continuous moral support throughout this project Without which, this journey will not be possible This thesis is dedicated to my dearest children, Kai-Xinn, Yi-Xinn and Yi-Kai I hope that one day, they will learn to discover and appreciate the beauty of physics ii Contents Acknowledgment i Summary vi Background and motivation 1.1 Equilibrium versus nonequilibrium fluctuations 1.2 Chaotic fluctuations 1.3 A chaotic kicked particle model 1.3.1 Generalized kicked particle map 1.3.2 Tchebyscheff maps as chaos fluctuation 11 1.4 Nonequilibrium transport 14 1.5 Motivations and outline of the thesis 15 Microscopic chaos, Gaussian diffusion processes and Brownian motion 18 2.1 Introduction 18 2.2 The Beck map 20 2.3 Statistical physics 21 2.3.1 Equipartition theorem 22 2.3.2 Mean square displacement and Einstein’s diffusion 22 2.3.3 Green-Kubo relation 24 2.3.4 Power spectrum 27 2.3.5 Discussion 27 iii 2.4 Gaussian diffusion process from Beck map 29 2.4.1 Skewness and kurtosis 30 2.4.2 Weak convergence to the Gaussian diffusion process 33 The Feynman’s graph approach 36 2.5 Non-Brownian feature of the position process 39 2.6 Summary of main results 42 2.4.3 Brownian ratchets from chaotic fluctuations 43 3.1 Introduction 43 3.2 A strongly damped kicked particle model 45 3.2.1 Chaotic kicked particle model in a constant force field 45 3.2.2 Quasi-stationary kicked particle map 47 3.3 The Perron-Frobenius approach to nonequilibrium transport equation 48 3.4 Chaotic fluctuations from double symmetric maps 54 3.5 Directed current from periodic potential 56 3.6 Tilting ratchet with symmetric cosine potential and asymmetric fluctuations generated by G(2) map 60 3.7 Comparison between theory and numerical simulations 63 3.8 Summary of main results 67 Statistical asymmetric and symmetric chaos-induced transport 69 4.1 Introduction 69 4.2 Chaotic kicked particle in a reflective and symmetric potential field 72 4.2.1 A harmonic potential 73 4.2.2 A bistable potential 75 4.3 Chaos-induced escape over a potential barrier 77 4.4 Chaotic resonance 86 4.4.1 Chaotic kicked particle model with sinusoidal force of slow variation 4.4.2 87 Two-state model of chaotic resonance from G(2) map 89 iv 4.5 Summary of main results Conclusion A Calculation of ( 97 98 p n i=1 ci sFi ) N with p = 1, 2, 3, 102 B Summing the Feynman’s graphs 108 C A lemma on separation of distribution functions 114 Bibliography 115 v Summary In this thesis, we present a mathematical theory on the transport of mesoscopic particle under the action of a class of nonequilibrium chaotic fluctuations By considering a perturbative Perron-Frobenius approach, we arrive at a transport equation for our strongly damped particle in a potential field This transport equation has the form of an inhomogeneous Smoluchowski equation with a source term We have identified the source term to be associated with the statistical asymmetry of the chaotic fluctuations, which is the driving force behind various nonequilibrium transport phenomena In particular, the theory has enabled us to give a quantitative account for the occurrence of directed current in a spatially periodic potential, the desymmetrization of particle distributions within infinite symmetric potential wells, and the enhancement or suppression of the Kramers escape rate In addition to giving first-order analytical expressions to these phenomena, which have been duly verified through numerical simulations, the theory has also provided further insights to the correspondence between chaotic and stochastic resonance Interestingly, by turning towards the zeroth-order limit, our theory shows that the particle distribution obeys the stochastic Smoluchowski equation even though the fluctuating force is deterministic In the situation of free field, the relation between determinism and stochasticity seems to extend beyond the leading order In this context, we have found interesting connection of the particle dynamics with statistical mechanics, and prove the surprising result that a non-Ornstein-Uhlenbeck deterministic process can generate a stochastic Gaussian diffusion process by means of the Salem-Zygmund theorem and the approach of Feynman’s graph vi Chapter Background and motivation 1.1 Equilibrium versus nonequilibrium fluctuations The second law of thermodynamics prohibits usable work to be extracted from equilibrium fluctuations It is impossible to create a Maxwell’s demon mechanism within a thermodynamically equilibrium system However, a classic thought experiment based on a system of miniature ratchet and pawl, suggested by Richard Feynman in his famous Lectures on Physics [1], has almost revived such a Maxwell’s demon Feynman’s contrivance consists of a ratchet with a spring loaded pawl at one end, while a paddle wheel at the other, and they are connected by an axle This device, also known as the Feynman ratchet, is then immersed in an equilibrium heat bath On first look, one imagines that as the paddle wheel is kicked equally in both directions by the thermal fluctuations, the ratchet can perceivably turn only in the forward direction because it is prevented from moving backward by the action of the pawl In seeming contradiction to Carnot’s principle, a perpetual motion machine of the second kind has been obtained! But that is not so The spring, which is attached to the pawl, is subject to the same thermal fluctuations, such that it lifts the pawl away from the ratchet occasionally As a consequence, the ratchet moves forward and backward with equal frequency, and the second law of thermodynamics is saved Although the non-violation of the second law seems to result from a proper microscopic interpretation of the Feynman ratchet, it is in fact based on a deeper physical reason: the tumultuous molecular world faced by all parts of the contraption is under the same state of equilibrium fluctuation (The story would be very different if nonequilibrium fluctuation is encountered instead.) Equilibrium fluctuations are minute kicking forces that a Brownian particle experiences in a heat bath that is in a state of thermodynamic equilibrium The thermodynamic equilibrium state is a state of maximum entropy (most probable state), with the probability of an entropy change ∆S about this maximum given by Einstein’s formula [2]: P (∆S) = Z −1 e∆S/k , (1.1) where Z is the normalization constant and k the Boltzmann constant If we were to express ∆S in terms of a set of n independent variables that describe the deviation from this equilibrium state by αk , the concavity of the entropy necessitates the following quadratic form: ∆S = − gij αi αj , (1.2) i,j in which gij are appropriate coefficients while the negative sign emphasizes the fact that ∆S is a negative quantity Thus, the fluctuation at equilibrium obeys a multivariate Gaussian distribution function: P (α1 , α2 , · · · , αn ) =  det g exp − n (2πk) 2k n i,j=1  gij αi αj  , (1.3) with det g being the determinant of the matrix gij The interaction of such an equilibrium fluctuating force with a Brownian particle of momentum p is best described by the Langevin formulation: p(t) + V (x(t)) = −γp(t) + ξ(t) , ˙ (1.4) where x(t) is the coordinate of the particle in one-dimension and V (x) = dV (x)/dx The Langevin equation relates the deterministic and conservative part of the dynamics on the left-hand side with the effects of the thermal environment on the right-hand side More specifically, the thermal effect is due to a viscous damping with coefficient γ and a random fluctuating force ξ(t) The rapid fluctuation ξ(t) is generally assumed to be the Gaussian white noise1 This is a mathematical idealization, because in reality, the correlation time of the physical fluctuation is expected to be finite though negligibly small in comparison with the other relevant time scales within the system [4] It is possible to arrive at the phenomenological model given by Eq (1.4) through microscopic modeling, which makes the physics more transparent This has been carried out by exploiting the fact that the Brownian particle is more massive than the surrounding fluid molecules, with the result that the heavier particle is relatively slow compared to the fast motion of the much lighter molecules The method of elimination of fast variables then leads to Eq (1.4), with the following result in the lowest order [5]: ξ(t) = (1.5) ξ(t)ξ(s) = 2mγkT δ(t − s) , (1.6) and where m is the mass of the Brownian particle, T is the temperature of the heat bath and δ(t) is Dirac delta function Equation (1.6) is notably the fluctuationdissipation relation, which indicates that the viscous force and the fluctuating force are not independent As the Brownian particle moves within the heat bath of fluid molecules, the molecules adjust their distribution rapidly to the particle’s slower motion, but not instantaneously and the lag causes the damping force, which takes energy away from the particle On the other hand, ξ(t) captures the incessant collision of the fast molecules, and it is where energy is given back to the Brownian particle Such a dissipation and fluctuation mechanism has led to the view of γ as the coupling strength to the thermal environment This ab initio assumption of a clear-cut separation of time scales between a slow variable and the fast variables of the environment amounts to a neglect of the hydrodynamic modes of the fluid An interaction with a heat bath that is in thermodynamic This stochastic assumption leads to the conclusion that x(t) is continuous but not differentiable Mathematically, p(t) is ill-defined A more formal approach is to interpret the Langevin equation in ˙ the form of a stochastic integral equation [3] to xr n+1 N , where ci = − e−(n−i+1)γτ Equation (B.3) can be simplified by ignoring the condition that all the ji ’s are necessarily different in the summations, which incurs an error of order n−1 With r = 2s for the trivial double trees, this equation becomes s  (2s)!  n  (N ) cj + O n−1 E0 (2s) = 2−2s (4Dτ )s s! j=1 (B.4) In consequence, only the even-order moments contribute to the zeroth-order charac(N ) teristic function Γ0 (k) as follows: (N ) Γ0 (k) = ∞ (ik)r (N ) E0 (r) r=0 r!  ∞ s i2s k 2s −2s (2s)!  n  = (4Dτ )s cj s! s=0 (2s)! j=1  s n 1 −Dτ c2  k 2s = j s! s=0 j=1 ∞  = exp −Dτ = exp −βk where β = Dτ n j=1 cj j=1 , c2 k  j (B.5) With n n c2 j  n = j=1 j=1 = n− ≈ n − e−(n+1−j)γτ 2e−γτ (1 − e−nγτ ) e−2γτ (1 − e−2nγτ ) + − e−γτ − e−2γτ (B.6) (N ) for n → ∞, we get β = Dnτ and Γ0 (k) = exp (−Dnτ k ) Employing Eq (B.2), we obtain the zeroth-order probability distribution function (N ) P0 (xn+1 ) = √ x2 exp − n+1 4Dnτ 4πDnτ in the asymptotic limit n → ∞ 109 (B.7) (II) First-order contribution – O(n−1/2 ) Let us now determine the next order to the asymptotic contribution to the Gaussian diffusion process given in Eq (B.7) The contribution comes from forests with r = 2s + leaves, in which one of the tree is a fork, while the rest are trivial double binary trees These forests, which exists only for N = 2, are illustrated in Fig 2.7 By applying the ‘Feynman rules’ to forests of this form, we have   s−1 n (2s + 1)!  n (2) E1 (r) = (4Dτ ) (Dτ )s cj c2   c2  j j−1 2(s − 1)! j=2 j=1 +O n−1 (B.8) Notice that only the odd-order moments contribute to the first-order characteristic (2) function Γ1 (k) in this case, i.e., (2) Γ1 (k) = ∞ (ik)r (2) E1 (r) r=0 r!  ∞  n i2s+1 k 2s+1 s (2s + 1)!  (Dτ ) = (4Dτ ) cj cj−1  2(s − 1)! j=2 s=1 (2s + 1)!  × n j=1 s−1 c2  j (B.9) Equation (B.9) can be simplified The result is (2) Γ1 (k) = −ik (4Dτ ) β1 exp −βk , (B.10) where n c2 j β := Dτ j=1 ≈ Dnτ (B.11) and β1 := ≈ n Dτ cj c2 j−1 j=2 Dnτ 110 (B.12) as n → ∞ The Fourier transform of Eq (B.10) leads to (2) P1 (xn+1 ) = (4Dτ ) 2 β1 xn+1 xn+1 − 6β 8β √ x2 exp − n+1 4β 4πβ (B.13) From Eq (2.17), we know that xn+1 ∼ n1/2 Thus, at asymptotically large n, the prefactor of the Gaussian diffusive term of Eq (B.13) scales as (4Dτ ) Dnτ xn+1 x2 − 6Dnτ n+1 8D3 n3 τ 1 n · n2 · n ∼ n− ∼ n (B.14) This has led us to write the first-order probability distribution function in the form: x2 (2) P1 (xn+1 ) = O n− √ exp − n+1 4Dnτ 4πDnτ (B.15) (III) Second-order contribution – O(n−1 ) In addition to summing the appropriate set of forests, which are to be found in Ref [35], the second-order contribution needs to account for the error made by assuming the summation indices not differ during the calculation of the zeroth- and first-order characteristic functions The details of this process have been worked out in the same reference, with all necessary terms evaluated in a similar fashion through the help of the ‘Feynman rules’ as before In the following, we will employ these results to determine the second-order asymptotic contribution to the Gaussian diffusion process To begin, let us first evaluate the second-order characteristic function of N = 2: (2) Γ2 (k) = 4Dτ (β3 − β2 )k − β1 k exp −βk , (B.16) where β2 := ≈ n c4 Dτ j 16 j=1 Dnτ 16 111 (B.17) and n cj cj−1 cj−2 Dτ j=3 Dnτ β3 := ≈ (B.18) as n → ∞ Hence, (2) P2 (xn+1 ) = 4Dτ b0 + b1 xn+1 + b2 xn+1 + b3 x6 n+1 ×√ x2 exp − n+1 4β 4πβ , (B.19) with 3(β3 − β2 ) 15β1 − , 4β 16β 3(β3 − β2 ) 45β1 + , = − 4β 32β β3 − β2 15β1 = − , 16β 64β β1 = 128β b0 = b1 b2 b3 The results indicate that b0 ∼ b1 x2 ∼ b2 xn+1 ∼ b3 x6 ∼ O n−1 n+1 n+1 (B.20) Thus, in the asymptotic regime, (2) P2 (xn+1 ) = O n−1 √ x2 exp − n+1 4Dnτ 4πDnτ (B.21) For N = 3, (3) Γ2 (k) = 4Dτ β4 k − β2 k exp −βk , (B.22) where β4 := ≈ n Dτ cj c3 j−1 12 j=2 Dnτ 12 112 (B.23) as n → ∞ The Fourier transform of Eq (B.22) is (3) P2 (xn+1 ) = 12 4Dτ β4 − β2 12 − x2 + x4 n+1 16 β β β n+1 x2 ×√ exp − n+1 4β 4πβ (B.24) Noting that β4 − β2 β4 − β2 β4 − β2 ∼ xn+1 ∼ xn+1 ∼ O n−1 , β β β4 (B.25) we again arrive at the asymptotic result: (3) P2 (xn+1 ) = O n−1 √ x2 exp − n+1 4Dnτ 4πDnτ (B.26) Finally, for N ≥ 4, (N ≥4) Γ2 (k) = −4Dτ β2 k exp −βk , (B.27) which implies that (N ≥4) P2 4Dτ β2 12 12 − x2 + x4 n+1 16 β β β n+1 x2 ×√ exp − n+1 4β 4πβ (xn+1 ) = − (B.28) With β2 β2 β2 ∼ x2 ∼ xn+1 ∼ O n−1 n+1 β β β (B.29) as n → ∞, we have (N ≥4) P2 (xn+1 ) = O n−1 √ x2 exp − n+1 4Dnτ 4πDnτ (B.30) Combining Eqs (B.21), (B.26) and (B.30), the second-order probability distribution function is of the form: (N ) P2 x2 exp − n+1 (xn+1 ) = O n−1 √ 4Dnτ 4πDnτ 113 (B.31) Appendix C A lemma on separation of distribution functions In this appendix, we show that the solution of equation ¯ α(F , x, t) = α(F, x, t) , −1 (F ) |G (F )| ¯ F ∈G (C.1) with F being the ergodic trajectory of a chaotic map G, takes the separable form [31] α(F, x, t) = h(F )A(x, t) , (C.2) A(x, t) = (C.3) where dF α(F, x, t) To begin, it is essential to note that Eq (C.1) relates two identical functions with ¯ respect to the variables F and F , while the values of x and t are the same at both sides of the equation As the chaotic maps in consideration are ergodic, an invariant density h(F ) exists and is unique [96] It also satisfies 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