23966k - Instituto Carlos I - Universidad de Granada

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• JLL: Equations that are phenomenological presumably are used because they actually work. So real systems do behave according to those phenomenological equations. We also believe that liquids are made up of molecules. These molecules undergo microscopic dynamics, so a given microstate of the liquid which is basically under classical mechanics would be a state where you have positions and velocities. • GC: On that we agree completely. Let me try restate my question. I still would like to understand what your mathematical problem is because if you start from phenomenological equations I think is very difficult to give to them a deep justification because they are phenomenological, so sometime they are valid and some other time not. . . • JLL: I’m totally on your side. I do want to understand the origin of these phenomenological equations. And I think that to understand the origin I have to go to the microscopic dynamics. • GC: Absolutely. • JLL: That’s what I’m saying. Going back to your simulations, suppose you start from a non-uniform macrostate, so you pick a random point in this region of the phase space. You would see it evolving, let’s say to the more uniform. Now you start with something which is more distorted; you see after a while it goes to the previous initial macrostate. Assuming you do your molecular dynamics perfectly exact, so you have now a new point in the phase space which has the property, if you reverse all the velocities it will go backwards, as compared as if you pick simply a point at random here. You will find that the reverse of the velocities will still make it go over here. It will not change at all. • GC: OK. • JLL: So the point in Γ2 that you get from coming from Γ1 is very different with respect to velocity reversal from a point you pick at random in Γ2. Nevertheless the system behaves macroscopically in a similar way in the future, as you can predict phenomenologically or see in your computer, and that’s what I think to prove mathematically or to understand mathematically is an essential problem. • GC: Then I totally agree. Your problem is strictly macroscopic. You start from the phenomenological equations when you find that they are satisfied. Why is this, it is not your problem. Fine. • Lev Shchur (LS): I don’t completely agree with you that it is possible to check in molecular dynamics with any precision. Typically, in the picture you showed us, you have some typical time scale, and Lyapunov exponents. Suppose you can make several mappings of such a kind but at some point you really forgot the initial one because you need some exponential precision for that. You know, your derivatives should be exponentially fine in a way. • JLL: I don’t think Giovanni says that you can do it experimentally, with infinite precision. But at least in some simulations we have done on integer arithmetic we did not have that problem, and still have the same answer. And, after all, the real system, presumably, to the extent that you can make it isolated, which again is not perfect, you cannot make it isolated, but I don’t think anyone here doubts that if you make a truly isolated system and you did truly infinitely precise molecular dynamics, that you would get any different answer. • LS: Yes, because in your picture you have Lyapunov exponent which is somehow connected with the Kolmogorov-Sinai entropy, which is a different entropy which characterizes the topology of your domains. What is not understandable also from your lecture,
is which domain you keep in your mind. Maybe you speak about some special domain on which you define your system. • JLL: Let’s say for the simplest case I would think of a hard sphere system, one with a Lennard-Jones potential in a given box, conceptually totally isolated, with reflecting boundary conditions. That’s what I have in mind. As far as one understands, at least in certain regimes, the phenomenological equations would be just absolutely well. • Miguel Rubí (MR): I want to make a comment on the first point. The question you raised is very important, to know under what conditions concepts introduced for systems in equilibrium such as entropy, free energy or temperature, can also be used in systems away from equilibrium. I think that this is a crucial point. In my opinion, this question is very much related to the nature of the noise present in the system. The nature of the noise may sometimes justify or invalidate the use of equilibrium concepts in non-equilibrium situations. So, for the particular case of a Gaussian noise, which is common to many physical systems, one can extend the use of some of these concepts. When the noise is Gaussian, the stochastic variables vary in a small amount for short time intervals, so the process is very slow and the probability distribution undergoes a diffusion processes governed by a Fokker-Planck equation which is simply a diffusion equation, linear in the probability. In these cases you can establish a connection with the second law and use the concept of entropy. However, if the noise is not Gaussian one has to use other kinetic equations the form of which is different from a diffusion equation. In these cases there is not a clear connection between the dynamics and the second law. To obtain an entropy functional compatible with these kinetic equations is not obvious. An example is the case of a multiplicative noise or other noises such as Poisson or Shot noises. When the distribution function is non-Gaussian, we do not know how to define the temperature. Sometimes, one may define effective temperatures but these quantities are not robust, they depend on the particular situation you are considering, on the observables and the initial conditions. These temperatures are not temperatures in a thermodynamic sense. When the noise is Gaussian the temperature inferred from the variance of the distribution coincides with the thermodynamic temperature and everything fits perfectly well. I want to make a comment on the distinction between the Gaussian nature of the noise and the Gaussian form of the probability distribution function. In the example I discussed in my talk on the Brownian motion in a shear-flow, in a stationary flow, the noise is Gaussian because the particle is immersed in a heat bath. But if you solve the Smoluchowski equation for a nonlinear velocity profile, for example a Poiseuille flow, you don’t get a Gaussian form. A Gaussian noise for which the kinetic equation is of the Fokker-Planck type does not necessarily imply that the probability distribution be a Gaussian. In the case of a gentle Poisseuille flow in which the system is not far from equilibrium this distribution function is non-Gaussian. Since the noise is Gaussian you can establish a connection with the second law and use the concept of entropy. So I insist in the fact that the nature of the noise is a crucial point in establishing the link between thermodynamics and stochasticity. • JLL: You are talking about noise. Presumably, this noise you expect is really internal. I mean, you don’t think the system would behave differently if it was truly isolated and followed the laws of the dynamics of classical mechanics. Whatever noise comes out, I mean, from the dynamics. Of course you don’t want to follow the molecules, you substitute, you replace it by some kind of noise. But I’m asking the questions at a more
Page 1 and 2: NON-EQUILIBRIUM STATISTICAL PHYSICS
Page 3 and 4: ISBN 978-0-7354-0887-6 ISSN 0094-24
Page 5 and 6: To learn more about AIP Conference
Page 7 and 8: Editors Pedro L. Garrido Joaquín M
Page 9 and 10: On the approach to thermal equilibr
Page 11 and 12: Analytical study of hysteresis in t
Page 13 and 14: Fluctuations of the dissipated ener
Page 15 and 16: EDITORS' PREFACE This volume origin
Page 17 and 18: Nonequilibrium Statistical Physics
Page 19: The third issue is really, if you w
Page 23 and 24: Still, if I give you a number, pick
Page 25 and 26: sions from kinetic theory, even at
Page 27 and 28: the forces between them, then we wo
Page 29 and 30: • JLL: I think in general. . . Wh
Page 31 and 32: something some interactions that ap
Page 33 and 34: equations of motion. Working just o
Page 35 and 36: you have some other open problems w
Page 37 and 38: Three lectures: NEMD, SPAM, and sho
Page 39 and 40: E = 1/2 0.3 Fermi-Pasta-Ulam 0.2 0.
Page 41 and 42: Consider the simplest interesting c
Page 43 and 44: Nosé-Hoover mechanics opens up the
Page 45 and 46: FIGURE 5. A series of 200,000 Galto
Page 47 and 48: FIGURE 7. A Four Chamber viscous fl
Page 49 and 50: FIGURE 8. Rayleigh-Bénard problem,
Page 51 and 52: SPAM Algorithms and the Continuity
Page 53 and 54: FIGURE 10. Contours of average dens
Page 55 and 56: FIGURE 12. Water Column collapse fo
Page 57 and 58: FIGURE 14. Stationary shockwave in
Page 59 and 60: FIGURE 16. Snapshots near the begin
Page 61 and 62: FIGURE 17. Shock Thermal and Mechan
Page 63 and 64: Solving the time-dependent continuu
Page 65 and 66: time = 2 time = 4 time = 6 time = 1
Page 67 and 68: time = 2 time = 4 time = 6 time = 1
Page 69 and 70: Holland, Amsterdam, 1986, pp. 43-65
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λ0 0000000 1111111 0000000 1111111
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The Langevin equation for the overd
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for the Langevin equation (4). The
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The first term on the right hand si
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As an important application, based
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FIGURE 6. Sketch of protein unfoldi
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Detailed fluctuation theorem In a N
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1 . 5 1 . 0 0 . 5 v io la t io n in
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At first sight, one might not have
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of internal states of the machine o
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Hydrodynamics from dynamical non-eq
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conclusions. THEORETICAL BACKGROUND
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= exp[−β(H0(Γ) + k 2 ( ˆF(r)
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y the Blue Moon ensemble (see Refs.
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ent cells as , b three servoir n x
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(a) (b) FIGURE FIG. 5. 3. Setting L
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simulation box along the χ-th Cart
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!"!"!#$%&'# !"!"!(#$%&'# FIGURE 6.
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produce the phenomenon of pushing u
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11. R. Kubo, J. Phys. Soc. Jpn. 12,
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According to the equipartition law,
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FIGURE 1. Control parameter and con
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FIGURE 2. The bead in the optical t
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The unfolding of the hairpin is rev
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FIGURE 5. The Crooks fluctuation re
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FIGURE 6. FDC versus FEC. (a) Exper
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FIGURE 8. The generalized Crooks fl
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Universality in equilibrium and awa
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UNIVERSALITY IN NON-EQUILIBRIUM Let
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From this perspective, does it make
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(a) (b) (c) (d) σ=2.30 σ=2.38 σ=
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to the coarse-graining process. How
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ford, 1990; G. Parisi, Statistical
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Fourier law, phase transitions, and
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FIGURE 1. a β (m) is flat in [−m
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which shows that in the stationary
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FREE ENERGY FUNCTIONALS AND LYAPUNO
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If we use instead the Ginzburg-Land
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REFERENCES 1. A. De Masi, E. Presut
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scale. The theory, mesoscopic non-e
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where L(γ,P(γ)) is an Onsager coe
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To illustrate explicitly the influe
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which can also be expressed as J =
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CONCLUSIONS The classical way to st
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Noise-induced transitions vs. noise
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Alternatively, for fixed µ > 0 one
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expressed in terms of the variable
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the dynamics: for larger noise inte
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m(t) m(t) m(t) m(t) m(t) 4 2 0 -2 -
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On the approach to thermal equilibr
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We assume that there is one dominan
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we thus have that |ψt〉〈ψt| =
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To better appreciate the significan
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with a multitude of systems, all wi
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hypothesis [2], in which the thermo
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NON-EQUILIBRIUM THERMODYNAMIC POTEN
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and composition, the question about
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3. D. Y. Tzou, Macro-to-microscale
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Dynamical processes defined on netw
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For the marginal case s = 2, where
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may occur, which cause power-law dy
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Stationary points approach to therm
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is an insignificant restriction, si
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FIGURE 2. Sketch of stationary poin
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dimensions and height comparable to
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FIGURE 2. Sequence of top view conf
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ACKNOWLEDGMENTS We would like to th
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Local properties at equilibrium can
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✻ β −1 µ0 µ1 µ2 µ3 J compl
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On the role of Galilean invariance
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with δ ± λ j ≡ 2νa (h j±1
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ACKNOWLEDGMENTS HSW acknowledges fi
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where 〈...〉eq is an average ove
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the Heisenberg chain is incorporate
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Computing LDFs from scratch, starti
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which expresses the locally-Gaussia
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allowed because of (i). Hence for t
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G(J x ,J y ) 0 -1 -2 -3 -4 -5 -6 -4
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dependent optimal profiles (probabl
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FIGURE 1. a) SNR vs γ, for εc −
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can set up a reference frame S ′
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A Φ(x) =ξ− B Φ(x) =ξ+ Φ(x) =
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Dynamical behavior of heat conducti
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Fast transients in mesoscopic syste
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ACKNOWLEDGMENTS This research was p
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FIGURE 1. (a) Three types of decay
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To study the heat flow, we write th
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the time-reversal invariance of the
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FIGURE 1. Averaged stochastic traje
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Strong ratchet effects for heteroge
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Stochastic protein production and t
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Creating the conditions of anomalou
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Cluster size distribution in Gaussi
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A non-equilibrium potential functio
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Quasi-stationary states and a class
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Fluctuation relations and fluctuati
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Why are so many networks disassorta
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Analytical study of hysteresis in t
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Queues on narrow roads and in airpl
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Fluctuations out of equilibrium V.V
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Long-range interacting systems and
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Irreversibility of the renormalizat
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Classical systems: moments, continu
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ABSTRACTS OF SELECTED CONTRIBUTIONS
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Nonlinear Boltzmann equation for th
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The Liouville equation and BBGKY hi
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Experimental densities of binary mi
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New insights in a 2-D hard disk sys
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Velocity-velocity correlation funct
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Analysis of ship maneuvering data f
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Thermally activated escape far from
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Space-time phase transitions in the
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Fluctuations of the dissipated ener
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Breeding gravitational lenses J. Li
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A formula on the pressure for a set
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Noninteracting classical spins in a
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Current fluctuations in a two dimen
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Violation of fluctuation-dissipatio
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Nonequilibrium thermodynamics of si
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The non-equilibrium and energetic c
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Dynamical systems approach to the s
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Akiyama, Ryo Japan Álvarez-Estrada
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A Akiyama, Ryo 261, 295 Albano, Eze
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P Pérez-Espigares, C. 202, 268, 28
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23966k - Instituto Carlos I - Universidad de Granada

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