Deterministic model of battery, uphill currents and non-equilibrium phase transitions
Emilio N.M. Cirillo, Matteo Colangeli, Omar Richardson, Lamberto Rondoni
DDeterministic model of battery, uphill currents and non–equilibrium phase transitions
Emilio N. M. Cirillo, ∗ Matteo Colangeli, † Omar Richardson, ‡ and Lamberto Rondoni
4, 5, § Dipartimento di Scienze di Base e Applicate per l’Ingegneria,Sapienza Universit`a di Roma, Via A. Scarpa 16, I–00161, Roma, Italy. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica,Universit`a degli Studi dell’Aquila, Via Vetoio, 67100 L’Aquila, Italy. Department of Mathematics and Computer Science, Karlstad University, Sweden. Dipartimento di Scienze Matematiche, Politecnico di Torino,Corso Duca degli Abruzzi 24, I–10129 Torino, Italy INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
We consider point particles in a table made of two circular cavities connected by two rectangularchannels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce–back mechanism acts when the number of particles flowing in one direction exceeds a given threshold T . In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. Thesecond channel is divided in two halves parallel to the first, but located in the opposite sides of thecavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities,of the channels and of T , non–equilibrium phase transitions take place in the N → ∞ limit. Thisinduces a stationary current in the circuit, thus modeling a kind of battery, although our model isdeterministic, conservative, and time reversal invariant. PACS numbers: 64.60.Bd, 05.45.-a, 05.40.-aKeywords: Billiards; Ehrenfest urns; ergodicity; non–equilibrium phase transitions.
I. INTRODUCTION
The nature of non–equilibrium phenomena is diverseand rich, and a theory encompassing them is still in themaking [1–4]. Such a task requires, in particular, under-standing the coupling with external agents or reservoirsthat may locally allow the condition of detailed balanceor its violation [5–9]. In this work we provide numericalresults and a theory explaining the onset of stationarycurrents in deterministic conservative reversible systemsmade of N point particles. Such currents are generatedby non–equilibrium phase transitions, that result in adeterministic model of battery, which is phase space vol-umes preserving and time reversal invariant [10]. Flowsand oscillations produced by this mechanism resemblethose observed in biological systems or chemical reac-tions, cf. Refs. [11, 12] for classical and quantum oscil-lators, and Ref. [13] for experiments on time crystals.In particular, our single component deterministic modelshows a realization of uphill currents , i.e. currents oppos-ing the driving fields, thus providing an instance of the ∗ [email protected] † [email protected]; ORCID 0000-0002-7424-7888 ‡ [email protected] § [email protected]; ORCID 0000-0002-4223-6279 so-called negative absolute mobility [14–16]. A theoreti-cal description of uphill diffusions was given in [17, 18]for stochastic spin models coupled to external reservoirs;moreover, in [19, Sec.4.5] uphill currents were also ob-tained from the scaling limit of inhomogeneous randomwalks on a lattice. A non–equilibrium phase transitionoccurring in a deterministic particle system was recentlyobserved in a model with two cavities connected by asingle channel, allowing no stationary currents [20]. Thetransition amounts to switching from a homogeneousstate, in which approximatively the same number of par-ticles lies in each urn, to an inhomogeneous state in whichalmost all particles gather in a single urn. The modelstudied in [20] was also amenable to a stochastic inter-pretation, in terms of time-dependent Markov chains. Inthis work, we investigate the nature of the steady state ina two-urns model equipped also with a second channel,that permits to close the system as in a circuit. The mainquestion we address here is two–fold. First, we shed lighton the existence of non–equilibrium phase transitions forthe circuit model, in which the second channel is designedto contrast the formation of particle gradients betweenthe urns. Furthermore, we also discuss the emergence ofstationary currents, flowing through the circuit and sus-tained by the phase transitions. We shall thus unveil anon–trivial phase diagram for our model, revealing that a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b phase transitions indeed occur in certain regions of theparameter space and are always followed by stationarycurrents.The work is organized as follows. In Sec. II we in-troduce our model and also present the numerical resultsof our deterministic dynamics. In Sec. III we tacklethe theoretical investigation of the model by means ofprobabilistic arguments, and also compare the theoreti-cal prediction with the numerical results. We also high-light strength and limitations of the probabilistic model.More details on the probabilistic derivation are deferredto the Appendix . Finally, conclusions are drawn in Sec.IV. II. THE MODEL
Our model consists of N point particles that move instraight lines with speed v = 1, and collide elasticallywith hard walls. Hence, from collision to collision, theparticles follow these equations of motion: ˙q = p and ˙p = . Therefore, their speed is preserved while theirvelocity is reflected with respect to the normal to theboundary of the table, at the collision point. The bil-liard table is made of two circular urns of radius r , con-nected by two rectangular channels of widths w, w (cid:48) andlengths (cid:96), (cid:96) (cid:48) , called, respectively, first and second chan-nel, cf. Fig. 1. The two urns will be referred to, in thesequel, as urn 1 and urn 2, respectively. The first channelis divided in two parts, each of length (cid:96)/
2, called gates .Periodic boundary conditions are imposed by letting thesecond channel close the table on a circuit. This consti- r ℓ /2 ww ′ ℓ ′ /2 FIG. 1. The billiard table: the N point particles are repre-sented as small disks and velocities are represented by arrows.The two grey shaded regions, in the first channel, are thegates in which the bounce–back mechanism separately acts.The horizontal component of the velocity of the particles con-tained in one gate and moving toward the other is reversedwhenever their number is larger than the prescribed thresholdvalue T . tutes an ergodic billiard [21, 22]. We now add the bounce–back mechanism in the first channel: when the numberof particles in one gate, that are moving toward the othergate, exceeds a threshold T , the horizontal component ofthe velocity of those particles is reversed. The particlescoming from the other gate are unaffected by this mech-anism and continue their motion. Although this dynam-ics is deterministic, time reversible and phase volumespreserving, it can produce non–equilibrium phase transi-tions, because the bounce–back mechanism implementsa sort of negative feedback, that promotes the onset of anon–equilibrium steady state.For T ≥ N , the usual ergodic billiard dynamics is re-alized [21, 23]. Thus, for large N , the vast majority oftime is spent in a state in which approximately the samenumber of particles resides in each urn. That state, likeany other, is abandoned to reach still other states, withfrequency given by the ratio of the respective phase spacevolumes [24], therefore no state is strictly stable. Nev-ertheless, the lifetime of the homogeneous phase rapidlybecomes so long, with growing N , that such a lack of sta-bility turns physically irrelevant even at moderately large N , consistently with Boltzmann’s explanation of his H -theorem [25, 26]. For T < N , ergodicity also guaranteesthat, sooner or later, the threshold will be exceeded ina gate: as long as that event does not occur, the dy-namics is like the ergodic one, which eventually leads toa stationary homogeneous state. When the threshold isexceeded, the standard dynamics is interrupted by thebounce–back. As evidenced in [20], for large N and suf-ficiently small T /N , a larger concentration of particles inone urn leads to an increased frequency of activation ofthe bounce–back mechanism in the adjacent gate, whileparticles can flow in from the other urn, incrementing theeffect. As a consequence, one urn gets depleted of parti-cles, while the other urn increases its population, until asteady state is reached in which the flow of particles perunit time in the two directions equalize. In this scenario,a microscopic fluctuation suffices to trigger the transi-tion, even when starting from the homogeneous phase.In this work, the phenomenology is much richer: the sec-ond channel allows particles to flow freely, contrasting thetrend toward inhomogeneous states. Both homogeneousand inhomogeneous states can thus be realized, and thelatter support stationary self sustained currents, as in abattery.Each half of the table is now made of one urn, of theadjacent gate and of the adjacent semi–channel of length (cid:96) (cid:48) / N and N be the number of particles in the two halves, with N = N + N , we define the mass spread by χ = | N − N | /N (1)For simplicity, we define the net current by taking the ab-solute value of the difference of the number of particlescoming from opposite directions and crossing the verticalline separating the two gates, and by then dividing thisquantity by the elapsed time t [27]. Namely, let n ( t )and n ( t ) denote the number of particles that cross, dur-ing the time interval [0, t], the vertical line separating thetwo gates in the direction from urn 1 to urn 2 and in theopposite direction, respectively. The net current flowingin a given channel is thus given by the ratio | n ( t ) − n ( t ) | t (2)In the large t limit, such a discretely defined current set-tles to a stationary value related to the asymptotic bil-liard current. Due to the symmetry of the model, it isirrelevant whether the net current is positive or negative.The model has been simulated as follows. Our numer-ical algorithm updates, at each time step, the position ofall particles by moving them along straight lines, in thedirection of their velocities, over a distance v δt . Here δt denotes the time interval between two consecutive colli-sions of the particles with the physical boundaries of thetable and also with the fictitious vertical lines markingeither the boundary of each gate with the adjacent urn orthe junction between the two gates in the first channel.Elastic reflections at the boundaries of the billiard tableare implemented . The initial datum is chosen by fixingarbitrarily the number of particles inside each urn andselecting their positions and velocities at random withuniform distribution. Nevertheless, the values attainedin the steady state by the observables in Eqs. (1) and(2) resulted insensitive to the initial datum. In particu-lar, in all our simulations we verified that, for any valueof the parameters of the model, the same stationary val-ues of the mass spread and the net current are numeri-cally reached by starting both from χ (0) = 0 and from χ (0) = 1. The direction of the inward normal at the junction points be-tween the urns and the channels depends on the origin of thecolliding particle. Namely, a junction point is considered as be-longing to the urn or to the channel if the incoming particle isoriginally located, respectively, in the urn or in the channel. T / N w Mass spread ( (0) = 0.0) T / N w Current ( (0) = 0.0)
FIG. 2. Stationary values of the mass spread χ (left panel)and net current (right panel) for N = 10 , w = 0 . r = (cid:96) = (cid:96) (cid:48) = 1. The initial condition yields χ (0) = 0. In the leftpanel, the red pixels denote the homogeneous phase, whereasthe other pixels refer to the inhomogeneous phase. The whitelines mark the theoretical boundary between homogeneousand inhomogeneous steady states (see Eq. (6)). In Fig. 2, we have N = 10 , w = 0 . r = 1, (cid:96) = 1, (cid:96) (cid:48) = 1, and varying values of T /N and w (cid:48) . Initially, N/ χ and of thenet current are computed averaging post–collision data,namely, right after the collision of each particle with thewalls of the table. Simulations last 10 collisions, corre-sponding, on average, to 10 collisions per particle.For T /N in an interval that depends on w (cid:48) , and forsmall w and w (cid:48) , an inhomogeneous phase is observed to-gether with a stationary net current flowing in the circuit.In particular, Fig. 2 shows that for w (cid:48) below a certaincritical value, three different regimes are produced byvariations of T /N , corresponding to two non–equilibriumphase transitions. The agreement between the theoreti-cal solid white lines and the numerical results is imperfectbecause our theoretical calculations rely on probabilisticarguments, which are justified by the ergodic hypothe-sis, in the large N and small w, w (cid:48) limits. Hence, thetheory better describes the simulations if N grows. InRef. [20], where only the first channel is present, thegrowth of N produces only one interface between homo-geneous and inhomogeneous phases, that occurs at a spe-cific T /N value, for fixed geometrical parameters. Thesecond channel results, instead, in a more complex phe-nomenology, because the free motion of particles passingthrough it tends to equilibrate N and N . Therefore,two contrasting mechanisms are at work, and their inter-play, between w (cid:48) and T /N in particular, determines thesteady state.In fact, higher
T /N values make the bounce–backmechanism less likely, hence particles flow more easilythrough the first channel, while lower values make parti-cles more likely to bounce back. Flow through the secondchannel decreases or increases when w (cid:48) does.Figure 3 shows how two different phase transitions canbe encountered. For T = 2 (panel (a)), we are in theleft region of Fig. 2. Here, a homogeneous phase arises,because in the first channel particles frequently bounceback, making left and right flows vanish, but N and N equalize thanks to the second channel. For T = 18 (panel(c)), in the region to the right of Fig. 2, the first chan-nel allows particles to flow almost freely, with a 0 netcurrent, while just a few particles cross the second chan-nel, because much smaller than the first: w (cid:48) = w/
30. For T = 7 (panel (b)), we fall in the centre of Fig. 2, where aninhomogeneous state, characterized by N (cid:54) = N and bya stationary net current, persists longer than our simula-tions. The reason is that the bounce–back phenomenonis only partly mitigated by the flow through the secondchannel. As the net current in the first channel flows up-hill , i.e. against the population gradient, and downhill inthe second channel, the first channel acts like an emf . Thenumerical results illustrated in Fig. 3 agree with excellentnumerical accuracy with the theoretical prediction (blackdashed lines) discussed in Sec. III. Stationary uphill cur-rents in presence of a non–equilibrium phase transitionhave been previously observed for stochastic dynamics in[17, 18, 28]; an analogous behavior has also been identi-fied in [29] for locally perturbed zero–range processes. Inthese cases, uphill currents stem either from a local inho-mogeneity in the jump rates, or from the non–equilibriumcoupling of the bulk dynamics with external reservoirs,that breaks detailed balance. Our deterministic conser-vative dynamics accounts, instead, for the work done bythe bounce–back mechanism. This phenomenology canbe understood introducing a variation of the probabilis-tic model of Ref. [20], that agrees with our deterministicdynamics in the large N and small w, w (cid:48) limits. Detailscan be found in the Appendix. III. THEORETICAL DERIVATION
Using the uniformity of the distribution of the par-ticles and of their velocities, one first obtains that thenumber of particles in urn 1, say, entering channel 1 (or channel 2) per unit time is given by N wv/ ( πA ) (or by N w (cid:48) v/ ( πA )), where A = πr − r arcsin w r + w (cid:112) (2 r ) − w − r arcsin w (cid:48) r + w (cid:48) (cid:112) (2 r ) − w (cid:48) (3)The number of particles leaving urn 1 and successfullycrossing the first channel, per unit time, is then reducedby the bounce–back mechanism to N wvπA Γ[ T, N w(cid:96)/ (4 A )]( T − y, x ] = (cid:82) ∞ x t y − e − s d s , y > , being the Euler in-complete Γ function. Thus, in the probabilistic model,the number of particles leaving urn 1 per unit time, andreaching urn 2, minus those going from urn 2 to urn 1 isgiven by: η = N vπA (cid:104) w Γ[ T, λ ]( T − w (cid:48) (cid:105) − N vπA (cid:104) w Γ[ T, λ ]( T − w (cid:48) (cid:105) (5)where λ i = N i w(cid:96)/ (4 A ), for i = 1 ,
2. Correspondingly,a steady state implies η = 0, an equation that can besolved for N . From Eq. (2) it is immediately seen thatthe condition of stationarity amounts to the equality be-tween the net current flowing uphill in the first channeland the net current flowing downhill in the second chan-nel, as in a circuit. Inspection shows that N = N/ η = 0 and also that, for certain param-eters values, η changes sign in intervals not containing N/
2. Given its continuity, in those cases in which η hasmore than one zero in [ N/ , N ], one may ask which of thesteady states of the probabilistic model is stable. Giventhe smoothness of η , the linear stability is given by thesign of ( ∂η/∂N ): if positive the steady state is unstable,if negative it is stable. The points at which this deriva-tive vanishes delimit the domains of stability of differentsteady states; hence, as a definition of the theoreticaltransition line, we shall consider the locus of points suchthat ∂η∂N (cid:12)(cid:12)(cid:12)(cid:12) N = N/ = 0 . (6)In other words, we collect the points where the homoge-neous solution of the equation η = 0 becomes unstable.The stability criterion based on the derivative of η is illustrated in Fig. 3. The inset in the panel (a) ofFig. 3 shows two stable steady states for the probabilis-tic model, but only the homogeneous one is actually ob-served in the simulations. This is in accord with the ini-tial condition being homogeneous. Possible departures -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5000 10000 15000 20000 25000 c u rr en t t -1-0.8-0.6-0.4-0.2 0 0.2 0.5 0.6 0.7 0.8 0.9 1 η N /N (a) c u rr en t t -1 0 1 2 3 4 0.5 0.6 0.7 0.8 0.9 1 η N /N (b) c u rr en t t -4-3.5-3-2.5-2-1.5-1-0.5 0 0.5 0.5 0.6 0.7 0.8 0.9 1 η N /N (c) FIG. 3. Currents as functions of time, for r = (cid:96) = (cid:96) (cid:48) = 1, 0 .
01 = w (cid:48) (cid:28) w = 0 . N = 10 , and initial datum such that χ (0) = 0.For T = 2 (panel (a)) and T = 18 (panel (c)), a homogeneous steady state with zero net current is reached. For T = 7 (panel(b)), a stationary net current with N (cid:54) = N arises Disks (squares) represent numerically computed flows out of urn 1 (urn 2)in channel 1; triangles (diamonds) represent flows out of urn 1 (urn 2) in channel 2. Black dashed lines are the theoreticalvalues (obtained from the probabilistic model) of the stationary currents; solid red (blue) lines denote the net currents in thefirst (second) channel. The parameter η is plotted in the insets as a function of N /N , while the red disks indicate the stablestate reached by the deterministic dynamics. χ t FIG. 4. Mass spread with the same values of the parametersand of the threshold considered in Fig. 3: homogeneous andinhomogeneous states are both stable (blue curve, T = 2);only the inhomogeneous state is stable (red curve, T = 7);only the homogeneous state is stable (black curve, T = 18).The dotted and dashed lines indicate the theoretical values ofthe mass spread for the inhomogeneous states at T = 2 and T = 7, respectively. The initial datum is such that χ (0) = 1. from this state, with N = 10 , are expected to be ex-tremely rare. The panel (c) illustrates a case in whichthe homogeneous state is stable, is the only steady statefor the probabilistic model, and is reached with the de-terministic dynamics. In the panel (b), the homogeneousstate is unstable for the probabilistic model, and is notobserved in the simulations, despite initially N = N .While this shows that the probabilistic model describes quite well the currents in the deterministic dynamics,Fig. 4 shows that some difference remains at finite N .In particular it reports the behavior of the mass spread,for the same values of the parameters and of the thresh-old considered in Fig. 3, with an initial datum yielding χ (0) = 1. The red line shows the convergence to an inho-mogeneous steady state, also evidenced in the panel (b)of Fig. 3. The black line shows the estabishment of astable homogeneous state (see panel (c) of Fig. 3), inwhich the mass spread rapidly drops down around 0. Fi-nally, the blue line illustrates the case, highlighted in thepanel (a) of Fig. 3, in which a homogeneous and one in-homogeneous states are both stable for the probabilisticmodel. Starting at χ (0) = 1, the deterministic dynam-ics converges toward and lingers over the inhomogeneoussteady state, but then it moves away, eventually converg-ing to the homogeneous state. Therefore, for our finite N , the lifetime of the inhomogeneous state is short, thatof the homogeneous state is very long. That is why thelatter looks globally attracting for the deterministic dy-namics.In the left panel of Fig. 5, a horizontal slice of Fig. 2,net currents are plotted as functions of T /N for fixed w (cid:48) . Theoretical predictions and data from simulationsare compared, which reveals that for small values of w (cid:48) the match is good already at N = 10 . The right panel ofFig. 5 highlights the battery phenomenon, with the firstchannel generating the emf . The resulting net current, ne t c u rr en t T/N ne t c u rr en t χ FIG. 5. Net currents as a function of
T /N (left panel) andof χ (right panel) for N = 10 , w = 0 . r = (cid:96) = (cid:96) (cid:48) = 1, w (cid:48) = 0 . . . . η = 0, see Eq. (5). at fixed w (cid:48) , is linear with the mass spread χ , and itsslope increases with w (cid:48) , closely following the theoreticalprediction. The linearity is better realized for smaller w (cid:48) ,consistently with the conditions for the applicability ofthe probabilistic model to the deterministic dynamics. IV. CONCLUSIONS
We considered a deterministic conservative reversibleparticle system undergoing a non–equilibrium phasetransition, induced by a bounce–back mechanism in oneof the channels. Numerical simulations of the determinis-tic dynamics reveal the existence of a rich phase diagramin the plane w (cid:48) – T /N , which includes states with station-ary density gradients and stationary currents. Remark-ably, the relation between the mass spread and the netcurrent turns out being linear for small values of w (cid:48) , inagreement with the basic tenets of ohmic transport [30,Chapter 4]. The numerical simulations of the determinis-tic dynamics are also supported by a theoretical analysisbased on probabilistic arguments. The match betweennumerical and analytical results, that strictly requiresthe N → ∞ and w (cid:48) → w (cid:48) . Interestingly, the regime in which the probabilisticmodel may be meaningfully applied to the deterministicdynamics is relatively easy to achieve in practice. Somerelevant open questions still lay ahead; one, in particu-lar, concerns the existence of phase transitions and sta-tionary currents when considering different geometries ofthe channels and/or of the cavities, or by adding long–range particle interactions. Further challenging mathe-matical questions concern the investigation of the ther-modynamic limit of our model, the relaxation of the par- ticle system toward a nonequilibrium steady state, whichcould even exhibit anomalous behavior [31], and applica-tions to the modelling of physical and chemical kinetics. Acknowledgements.
ENMC and MC thank KarlstadUniversity for its kind hospitality. OR is grateful tothe Sapienza Universit`a di Roma and the University ofL’Aquila for their kind hospitality and thankfully ac-knowledges partial financial support of the GS Magnus-sons fond. LR has been partially supported by Ministerodell’Istruzione, dell’Universit`a e della Ricerca (MIUR)Grant No. E11G18000350001 “Dipartimenti di Eccel-lenza 2018- 2022”. The authors are grateful to the Labo-ratorio di Calcolo of SBAI, Sapienza Universit`a di Roma.
Appendix: Probabilistic derivation of stationarycurrents
A geometrical argument shows that given n particles uni-formly distributed in an urn of area A , having fixed speed v and such that the direction of their velocities is uni-formly distributed on [0 , π ], the typical number of par-ticles that, in a small time interval of length δ >
0, leavethe urn and enter a gate of width w and length (cid:96)/
2, isgiven by2 (cid:16) n πA (cid:17) w (cid:90) vδ arccos (cid:16) − xvδ (cid:17) dx = n wvδπA (A.1)We call p δ = wvδ/ ( πA ) the probability that one particlein the urn enters the gate in the time interval δ , and alsocall τ = (cid:96)π/ (4 v ) the typical time needed for a particleto cross it. We then introduce a partition of the timeinterval τ into segments of length δ .The probability that s particles enter the gate in thetime τ obeys, in the δ → λ s s ! e − λ with λ = n w(cid:96) A , (A.2)where n is assumed to be so large not to be sensiblyaffected by the number of particles entering the gate.Therefore, the probability that at most T particles enterthe gate in the time interval τ reads P τ = T (cid:88) s =0 λ s s ! e − λ = Γ( T + 1 , λ ) T ! , (A.3)Γ[ y, x ] = (cid:82) ∞ x t y − e − s d s , y > , being the Euler incom-plete Γ function.We now consider a larger time scale of order t , andintroduce a coarser partition of the latter into segments of r ℓ /2 w FIG. 6. The billiard table considered in the numerical simu-lations. c u rr en t T/N c u rr en t T/N
FIG. 7. Stationary currents as functions of
T /N , with N =500 and N = 1000 (left and right panel, respectively), for r = (cid:96) = 1, and w = 0 . . . N = 500 and N = 1000respectively, obtained from Eq. (A.4). length τ . We have that P τ t/τ corresponds to the typicalnumber of events such that, in each time interval τ , thereare at most T particles in the gate. On the other hand,the typical number of particles entering the gate in thetime τ , conditioned to the fact that their number be atmost T , is equal to λT Γ( T, λ ) / Γ( T + 1 , λ ). Hence, wefind that λτ Γ( T, λ )( T − t = n wvπA Γ( T, λ )( T − t (A.4)yields the typical number of particles that, up to time t , successfully exit the gate without being bounced–backby the threshold mechanism [20].In order to validate the theoretical derivation of thecurrents, based on these probabilistic arguments, we nu-merically implemented a deterministic dynamics similarto the one used in the paper. We considered a billiard ta-ble made of a single urn and a semi–channel, whose rightvertical boundary is an elastically reflecting wall, see Fig.6. Inside the semi–channel, a bounce–back mechanismworks as described in the paper. Positions and velocities r e l a t i v e d i ff e r en c e T/N r e l a t i v e d i ff e r en c e T/N
FIG. 8. Difference between the theoretical and the numericalvalues of the currents, divided by the corresponding theoreti-cal values, as a function of
T /N with N = 500 and N = 1000(left and right panel, respectively), with the same values ofthe parameters of Fig. 7. s pe c i f i c c u rr en t T/N
FIG. 9. Stationary currents, rescaled by N , as functions of T /N , with N = 500 and N = 1000 (open and and filled sym-bols, respectively), with the same values of the parameters ofFig. 7. Dotted and dashed lines correspond to the theoret-ical values of the current rescaled by N , with N = 500 and N = 1000, respectively. of the n particles are initially taken at random, inside theurn, with uniform distribution.Such dynamics guarantees that the number of particlesin the urn is approximately constant, as it equals n minusthe (fluctuating) number of particles in the semi–channel.This, hence, permits a direct check of Eq. (A.4).The stationary current departing from the urn is de-fined as the long time limit of the ratio of the numberof collisions against the right vertical boundary of thesemi–channel to the elapsed time [27].Figure 7 shows, in particular, the comparison betweenthe theoretical expression of the current, from Eq. (A.4),and the numerical values of the stationary current ob-tained from the simulation of the deterministic dynam-ics. The discrepancy between the theoretical and the nu-merical values of the current is highlighted in Fig. 8. Asvisible from Figs. 7 and 8, the agreement between theoryand simulations improves for decreasing values of w andfor growing values of T /N : namely, when the ergodicity of the billiard dynamics is restored.Finally, Fig. 9 highlights the behavior of the specificcurrent , i.e., the stationary current divided by N , fordifferent values of N . [1] R. Kubo, M. Toda, N. Hashitsume, Statisti-cal Physics II. Nonequilibrium Statistical Mechanics ,Springer-Verlag, Berlin (1991).[2] G.P. Morriss, D.J. Evans,
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