Appendix: A Simple Numerical Toy Model Various elementary models which are intended to illustrate statistical meth- ods in physics can be found in the literature. Well known are the urn model (Ehrenfest and Ehrenfest 1911) and the ring model (Kac 1959). Both are based on stochastic dynamical assumptions, applied in a given time direc- tion. The model to be discussed below is deterministic, and based on special initial conditions. Its main purpose is to illustrate the concept of a Zwanzig projection by means of the simple example of spatial coarse-graining. This coarse-graining is used for the definition of entropy, and, in a second step, also to construct a master equation. However, it then fails to represent a good approximation, thus demonstrating the importance of dynamical properties of a Zwanzig projection if it is to be useful. The model is defined by a swarm of nP free particles (i =1,...,nP), mov- ing with fixed velocities vi = v0 +∆vi on a ‘ring’ (a periodic interval) divided into nS ‘unit cells’ (j =1,...,nS), where the ∆vi’s form a homogeneous ran- dom distribution in the range ±∆v/2. Coarse-graining is defined by averaging over cells. All particles are assumed to start from random positions in the cell j =1. The Zwanzig concept of relevance is furthermore assumed not to distin- guish between the (distinguishable) particles, thus establishing an ‘occupation number representation on unit cells’. Since momenta are conserved here, only the spatial distribution of particles has to be considered. If nj particles are in the j th cell, the entropy according to this Zwanzig projection is given by nS nj nj S = − ln . (6.19) n n j=1 P P max It vanishes in the initial state, while its maximum, S =lnnS, holds for equipartition, nj = nP/nS. With respect to this coarse-grained entropy, the nP−1 model has a statistical Poincar´erecurrencetimetPoincar´e = nS , while the relaxation time, required for the approach to equipartition, is only of order nS/∆v. It is so short, since this coarse-graining is not in any way robust. Other 204 Appendix: A Simple Numerical Toy Model equilibrium distributions could be constructed by using intervals of different lengths. ThefirstplotoftheMathematica notebook7 below (plot1) shows the en- tropy evolution for nP = 100 and nS = 20 during the first 2000 units of time. The relaxation time scale is of the order 1000 units. Only integer times are plotted in order to eliminate otherwise disturbing lattice effects of the model. At some later time (see plot2), the coarse-grained representation no longer re- veals any information about the existence of a low-entropy state in the recent past, although this information must still exist, since motion can be reversed in this deterministic model. One can now simply enforce a two-time boundary condition (Sect. 5.3.3) by restricting all relative velocities ∆vi to integer multiples of nS divided by a large integer (e.g., by rounding them off at a certain figure). Consequences of this change are negligible for small and intermediate times, although the evolution is now exactly periodic (on an interval of 200 000 units in the chosen numerical example, which is a very small fraction of the statistical recurrence time). Relative entropy minima may occur at simple rational fractions of this interval (such as at 2/5 of 200 000 in plot3). When using the coarse-graining dynamically, one obtains a master equa- tion for mean occupation numbersn ¯j(t) in an ensemble of individual solutions. For v0 = 0 it would read [see (4.45)] d¯nj = λ(¯nj− +¯nj − 2¯nj) , (6.20) dt 1 +1 where λ (in this case given by ∆v/8) is the mean rate for particles to move by one unit in either direction. For v0 = 0, the result would hold in the center-of- mass frame. The lower smooth curve of plot4 shows the resulting monotonic increase of ensemble entropy, compared with the individual (fluctuating) so- lution of plot1. Evidently, the two curves agree only for about the first 2/∆v units of time. This demonstrates that this Zwanzig projection is not very appropriate for dynamical purposes, since some fine-grained (neglected) information remains dynamically relevant: the individually conserved velocities lead here to rele- vant correlations between position and velocity. The master equation, which does not distinguish between individual particles, allows them even to change their direction of motion. A slightly improved long-time approximation can therefore be obtained by assuming all particles to diffuse in only one direction (upper smooth curve of plot4) – equivalent to using a reference frame that moves with a velocity −∆v/2 in the center-of-mass frame. It still neglects dy- namically relevant correlations resulting from the fact that the most distant particles continue travelling fastest. These correlations remain dynamically relevant during relaxation, that is, until most particles have diffused once around the ring. This can be recognized in plot4. The master equation (6.20) may also represent an ensemble of individually stochastic (indeterministic) histories nj(t), described by a Langevin equation. 7 Programming aid by Erich Joos is acknowledged. Appendix: A Simple Numerical Toy Model 205 Notebook: A Toy Model 1. Definitions nS = 20; nP = 100; v0 = 1.; 'v = .02; shannon x_ := If x == 0, 0., N x Log x entropMax = Log nS N @ D @ @ @ DDD 2.995732274 @ Dêê poincareTime = nS^ nP 1 N 6.338253001 × 10128 H Lêê relaxTime = nS 'v 1000. ê 2. Initial Values SeedRandom 2 v = Table v0 + 'v Random .5 , nP ; @ D x0 = Table Random , nP ; @ H @D L 8 <D << "Statistics`DescriptiveStatistics`" @ @D 8 <D Mean x0 , Mean v , Variance v 0.5156437577, 0.9997682434, 0.00003356193642 8 @ D @ D @ D< 8 < 206 Appendix: A Simple Numerical Toy Model 3. Exact Model entropy t_ := Module cellnb, n, e , cellnb = Ceiling Mod vt+ x0, nS ; @ D A8 < iDo n i = Count cellnb, i , i, nS ; j j nS@ @ DD j j=1 shannon n j ke = N Log nP ccccccccccccccccccccccccccccccccccccccccccccccccccccccc nP @ @ D @ D 8 <D ⁄ @ @ DD y plot1 = ListPlot Table t, entropy t z , t, 50, 2000, 5 , A @ D E z E PlotJoined True, Frame True, PlotRangez 0, entropMax ; { @ @8 @ D< 8 <D 8 <D 2.5 2 1.5 1 0.5 0 500 1000 1500 2000 plot2 = ListPlot Table t, entropy t , t, 10000, 12000, 5 , PlotJoined True, Frame True, PlotRange 0, entropMax ; @ @8 @ D< 8 <D 8 <D 2.5 2 1.5 1 0.5 10000 10500 11000 11500 12000 Appendix: A Simple Numerical Toy Model 207 4. Two-Time Boundary Condition Floor 10000 v v = ccccccccccccccccccccccccccccccccccccccccccccc ; 10000 period = @10000 nSD 200000 plot3 = ListPlot Table t, entropy t , t, 79000, 81000, 5 , PlotJoined True, Frame True, PlotRange 0, entropMax ; @ @8 @ D< 8 <D 8 <D 2.5 2 1.5 1 0.5 79000 79500 80000 80500 81000 5. Master Equation O = 'v 8 0.0025 ê equList = Table n j ' t == O 2n j t + n Mod j, nS + 1 t + n Mod j 2, nS + 1 t , j, nS ; @ @ D @ D H @ D@ D iniList =@ Join@ n D1 0D@==D nP @, Table@ n j D0 ==D@0.,DL j,8 2, nS<D ; @8 @ D@ D < @ @ D@ D 8 <D D 208 Appendix: A Simple Numerical Toy Model nSolution = NDSolve Join equList, iniList , Table n j , j, nS , t, 0, 2000 ; nS @ @ j=1 shannonD n j@ t@ D entropyMaster t_ := N Log nP ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 8 <D 8 <D nP . nSolution First ⁄ @ @ D@ DD @ D @ @ DD plotMasterShort = ListPlot Table x, entropyMaster x , ê x, 0, 2000,êê 20 , PlotJoined > True, PlotRange > 0, entropMax , DisplayFunction > Identity ; @ @8 @ D< equList8 = Table n<Dj ' t == 4 O n j t 8 + n Mod j <2, nS + 1 t , j, nS ; D @ @ D @ D H @ D@ D nSolution@= NDSolve@ JoinD equList,D@ DL 8 iniList<D , Table n j , j, nS , t, 0, 2000 ; plotMasterLong = ListPlot@ @ Table x, entropyMasterD @ x@ D , 8x, 0,<D 2000,8 20 , PlotJoined<D > True, PlotRange > 0, entropMax , DisplayFunction > Identity ; @ @8 @ D< plot48 = Show plot1,<D plotMasterShort, plotMasterLong ; 8 < D @ D 2.5 2 1.5 1 0.5 0 500 1000 1500 2000 References References marked with (WZ) can also be found (in English translation if applica- ble) in Wheeler and Zurek 1983, those with (LR) in Leff and Rex 1990. Some of my own (p)reprints are available through www.zeh-hd.de. The page of this book on which the reference is quoted is added in square brackets. Aharonov, Y., Bergmann, P.G., and Lebowitz, J.L. (1964): Time symmetry in the quantum process of measurement. Phys. Rev. 134, B1410 – (WZ) – [131,201] Aharonov, Y., and Vaidman, L. (1991): Complete description of a quantum system at a given time. J. Phys. A24, 2315 – [126] Albrecht, A., (1992): Investigating decoherence in a simple system. Phys. Rev. D46, 5504 – [133] Albrecht, A. (1993): Following a “collapsing” wave function. Phys. Rev. D48, 3768 – [133] Alicki, R., and Lendi, K. (1987): Quantum Dynamical Semigroups and Applications (Springer) – [118] Anderson, E. (2006): Emergent semiclassical time in quantum gravity. E-print gr- qc/0611007 and gr-qc/0611008 – [193] Anglin, J.R., and Zurek, W.H. (1996): A precision test of decoherence. E-print quant-ph/9611049 – [110] Arndt, M., Nairz, O., Vos-Andreae, J., Keler, C., van der Zouw, G., and Zeilinger, A. (1999): Wave–particle duality of C60 molecules. Nature 401, 680 – [104] Arnol’d, V.I., and Avez, A. (1968): Ergodic Problems of Classical Mechanics (Ben- jamin) – [55] Arnowitt, R., Deser, S., and Misner, C.W. (1962): The dynamics of general relativ- ity.