Relaxation-speed crossover in anharmonic potentials
Jan Meibohm,1 Danilo Forastiere,1 Tunrayo Adeleke-Larodo,1 and Karel Proesmans1, 2 1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg 2Hasselt University, B-3590 Diepenbeek, Belgium In a recent paper [Phys. Rev. Lett. 125, 110602 (2020)], thermal relaxation was observed to occur faster from cold to hot (heating) than from hot to cold (cooling). Here we show that overdamped diffusion in anharmonic single-well potentials generically allows for both faster heating and faster cooling, depending on the initial temperatures and on the potential’s degree of anharmonicity. We draw a relaxation-speed phase diagram that localises the different behaviours in parameter space. In addition to faster-heating and faster-cooling regions, we identify a crossover region in the phase diagram, where heating is initially slower but asymptotically faster than cooling.
Many thermal relaxation processes in nature and in- mal relaxation to equilibrium can be substantially in- dustry occur out of equilibrium, and thus outside of creased by varying the anharmonicity of the potential. the realm of the quasistatic approximation. As a con- This should be testable in experiments and has potential sequence, non-equilibrium thermal relaxation gives rise applications in the optimisation of cooling strategies for to new phenomena, such as ergodicity breaking [1] or the small-scale systems [9]. Mpemba effect [2]. A better understanding of anoma- We now specify the problem. Consider two equilib- lous thermal relaxation in out-of-equilibrium systems is rium systems, otherwise identical, but at different tem- important, because it may allow us to use these non- peratures Tc < Th. We call the system at temperature Tc equilibrium phenomena to our advantage, for instance, cold and that at temperature Th hot. At time t = 0, both for increasing the rate of heating and cooling. systems experience an instantaneous temperature quench Although a complete understanding of anomalous to the same final temperature Tf , where Tc < Tf < Th. relaxation in macroscopic systems appears elusive at The relaxation of the two systems toward equilibrium present, much progress has been made recently in repro- is monitored by their non-equilibrium free-energy differ- ducing anomalous relaxation phenomena on mesoscopic ence [13], scales. This has led to several important results such as new theoretical [3–6] and experimental [7, 8] insights ∞ pi(x, t) i(t) = kBTf dx pi(x, t) log , (1) into the Mpemba effect, strategies to increase the rate at F pf (x) Z−∞ which systems can be cooled [9–11], and an information- theoretic bound on the speed of relaxation to equilib- with respect to the equilibrium distribution pf at final rium [12]. temperature Tf ; kB is the Boltzmann constant. The in- Within a setup closely related to, yet slightly different dex i in Eq. (1) takes the values c and h, and pc and ph from, the Mpemba effect, a recent study [13] reported an denote the (time-dependent) probability densities of the asymmetry in the rate at which systems heat up and cool initially cold and hot system, respectively. In order to quantitatively compare the distances (t) down. According to this study, and subsequent works Fi by other authors, heating occurs faster than cooling for from equilibrium as functions of time, the temperatures diffusive systems with harmonic potentials [13] and for Tc and Th at t = 0 are chosen so that c(0) = h(0) [13]. We call such a temperature quench “ F-equidistant”,F i.e., discrete-state two-level systems [14, 15]. On the other F hand, it was shown that this relaxation-asymmetry is at equal distance with respect to the distance measure non-generic for diffusion in potentials with multiple min- (1). A comparison between this setup and the Marko- ima [13] or in discrete-state systems with more than two vian Mpemba effect [3, 4] is made in the Supplemental states [14, 15]. However, it appears to be widely be- Material (SM), see Sec. II of [16]. lieved that the described effect is a general property of The specific measure (1) is used for two reasons. First, overdamped, diffusive systems with stable single-well po- i is a thermodynamic quantity for systems at equilib- F tentials [13–15]. rium and hence for t < 0 and in the limit t . Second, → ∞ In this Letter, we study the relaxation asymmetry for it remains well defined out of equilibrium and thus for all arXiv:2107.07894v1 [cond-mat.stat-mech] 16 Jul 2021 overdamped diffusion in anharmonic single-well poten- finite times t. tials. We show that, against common belief, these sys- In the long-time limit, both the cold and the hot sys- tem relax to equilibrium so that (t) and (t) tend to tems exhibit both behaviours, faster heating and faster Fc Fh cooling. Based on these results, we draw a phase di- zero asymptotically. The relative distance from equilib- agram locating the different regions of “faster heating” rium of the two systems is conveniently measured by the and “faster cooling” in parameter space. These two re- logarithmic ratio gions are separated by a crossover region where cooling (t) occurs faster at first, but heating overtakes at a finite (t) log Fh . (2) R ≡ (t) time. Our results suggest that the relative speed of ther- Fc 2
For overdamped diffusion in a harmonic potential, one 1 2 (a) (b) can prove that (t) > 0, i.e., (t) > (t) during the T 0 R Fh Fc h relaxation [13], i.e., heating occurs faster than cooling; 0 1 (t) < 0 corresponds to the opposite case, that of faster F T 0 Rcooling. Note also that (0) = 0 by definition of - c R F equidistance, c(0) = h(0). Hence, the momentary, 0 0 F F 0 0 relative distance from equilibrium is determined by the 0 Tc 1 2 Th 3 0 Tc 1 Th 2 sign of (t). R We study the evolution of (t) for overdamped dif- FIG. 1. (a) Free-energy difference 0 at time t = 0 for hot R F fusion in an anharmonic potential V (x). For simplicity, (red line) and cold (blue line) temperatures, h and c, re- T T we analyse the case of one spatial dimension and assume spectively. The -equidistance relation (8) is represented by F 2 α the grey lines. (b) h( c) (red line) and c( h) (blue line) for V (x) to be of the form V (x) = λx + k x , where we T T T T consider parameter values λ, k and α for which| | V is con- σ = 0, Eqs. (11). The grey lines and coloured labels indicate how these functions relate a temperature pair ( 0, 0). fining, V (x) as x . We move to a dimension- Tc Th less formulation→ ∞ by defining→ ±∞ a time scale τ and a length scale ` as Hence, after the -equidistant temperature quench Th 2/α 1/α F → 1 k − k − Tf and Tc Tf at t = 0, the evolution of the relative τ = , ` = . (3) distance from→ equilibrium, measured by (t) [Eq. (1)], µk T k T k T B f B f B f is a function of the parameters σ and α ofR the potential V (x) [Eq. (4)] and of the temperature ratios that enter Here, µ is the mobility. In the dimensionless coordinates, Ti times are measured in units of τ, lengths in units of ` and in the initial conditions (6). Prior to the temperature quench, the hot and cold sys- energies in units of kBTf . In particular, the transforma- ˜ ˜ tems are prepared at -equidistance so that their free- tion t t = t/τ, x x˜ = x/`, V (x) V = V/(kBTf ), F to dimensionless→ coordinates→ yields, after→ dropping the energy differences match. This condition implicitly re- tildes, the potential lates the hot and cold dimensionless temperatures, so that we can write ( ), with Tc Th V (x) = σx2 + x α , (4) | | h(0) = c(0) ( ) 0 . (8) F F Tc Th ≡ F with the dimensionless parameter σ = 2/α 2/α 1 Because has a single minimum at equilibrium where λk− (kBTf ) − . The parameter σ quantifies F 2 = 1 and = 0, there is always exactly one solution the importance of the harmonic term x compared T F α to Eq. (8) for which ( ) < . Figure 1(a) shows to the anharmonic term x . Specifically, σ is small Tc Th Th whenever: (1) λ is small,| i.e,| the harmonic coupling is schematically how the free-energy difference relates the weak, (2) k is large, corresponding to strong anharmonic different temperatures. coupling. In addition, one has the cases (3) 0 < α < 2 At t = 0, the formula for the dimensionless free energy difference 0 at equidistance [Eq. (1) in units of kBTf ] and small Tf , where the behaviour is dominated by the F (anharmonic) shape of the potential close to the origin, can be conveniently written as and (4) α > 2 and large Tf , i.e., the dynamics takes Z place in the anharmonic tails of the potential V (x). 0 = [1 + (1 )∂ 1 ] log T . (9) F − T T − Z The Fokker-Planck equation [17] that determines the 1 evolution of the probability density during the relaxation Here, = h when > 1 and = c when < reads, in the new coordinates, ∂ p (x, t) = p (x, t) with T T T T T T t i L i 1. Hence, in order to obtain the required -equidistant temperatures, we need to solve and invertF Eq. (9). This = ∂ [V 0(x) + ∂ ] , (5) L x x can be done analytically for σ = 0, where we find and initial conditions, 1 0 = [ (1 + log )] , (10) exp[ V (x)/ ] F α T − T p (x, 0) = − Ti . (6) i Z and by taking the inverse Ti c h The dimensionless temperature ratios i are either h = h = W 1 ce−T , c = W0 he−T . (11) T T T − − −T T − −T Th/Tf or c = Tc/Tf , depending on whether the initial T Here, W (x ), n = 1, 0 denotes Lambert (or product- temperature before the quench is Th or Tc. The constants n − Z in Eq. (6) are obtained from normalising the proba- log) function [18]. Note that the relations (11) are in- Ti dependent of α. This is because for σ = 0, α appears bility density. In the limit t , the densities pi(x, t) relax to the equilibrium distribution,→ ∞ only in the prefactor of 0 in Eq. (10), and thus drops out in Eq. (8). This ceasesF to be true for σ = 0, where exp[ V (x)] the implicit condition (8) must to be inverted6 numeri- pf (x) = − . (7) cally. Figure 1(b) shows ( ) (red line) and ( ) Z1 Th Tc Tc Th 3
(blue line) from Eqs. (11). For σ = 0 the curves remain (a) (b) 6 5 almost unchanged, but with a weak dependence on α and cooling faster cooling faster σ (not shown). 4 After preparing the hot and cold systems at - α equidistant temperatures, both systems are put in con-F 3 tact with the same heat bath at temperature = 1. At heating faster heating faster Tf 2 finite time t > 0, the probability densities pi(x, t) that 0.01 0.1 1 10 0.01 0.1 1 10 enter i(t) and thus (t), are obtained from the Fokker- T T PlanckF equation by R FIG. 2. Relaxation-speed phase diagram for long and short t pi(x, t) = eL pi(x, 0) . (12) times in the ( , α)-parameter space. (a) Phase diagram in the short-timeT limit. The dash-dotted line shows the critical In other words, in order to compute (t) we must eval- R line for σ = 0, the dashed lines show the critical line for uate the operator-exponential in Eq. (12). This can be σ = 0.5 and 1. The black arrows indicate how the critical line done in the short- and long-time limits, leading to pre- changes as σ is increased from zero. The dotted lines show cise asymptotic results for (t). As we show below, the the critical lines for σ = 0.1, 0.15, and 0.2. The white asymptotics of (t) provideR an excellent characterisation arrows indicate how the critical− line− changes− as σ is decreased of the dynamics,R also at finite t. from zero. (b) Phase diagram in the long-time limit. The For short times t 1, the logarithmic ratio (2) reads solid line shows the critical line for σ = 0, the dashed lines show the critical line for σ = 0.2 and 0.4, dotted lines show it for σ = 0.1, 0.2, and 0.3. The black and white arrows ˙h(0) ˙c(0) (t) ˙ (0)t = F − F t , (13) indicate how− the− critical line− changes, just as in Fig. 2(a). R ∼ R F0 where the dot denotes a time derivative and 0 is the initial free-energy difference given in Eq. (9).F Through Figure 2(a) shows the short-time phase diagram for ˙ (0) = ∞ dx∂ p (x, 0) log[p (x, 0)/p (x)], the short- σ = 0. Its left- and right-hand sides, coloured in blue Fi t i i f time evaluation−∞ of (t) depends on the time derivative and red, respectively, correspond to the -equidistant R R temperatures < 1 and > 1. The phaseF diagram ∂tp(x, 0), evaluated at t = 0. We obtain the latter by Tc Th expanding the exponential in Eq. (12). We write is separated into an upper and a lower part with differ- ent short-time behaviours. In the lower part, ˙ (0) > 0 R pi(x, t) pi(x, 0) + t pi(x, 0) , so that (t) is initially positive for all pairs ; heating ∼ L R Ti = [1 + t(1 )∂2]p (x, 0) , (14) is faster than cooling. In the upper part, cooling is ini- − Ti x i tially faster than heating. The two parts are separated which gives by a critical line (blue and red dash-dotted line) where ˙ (0) = 0 so that (t) vanishes to first order in time, ∂ p (x, 0) = (1 )∂2p (x, 0) . (15) R 2 R t i − Ti x i (t) (t ). For σ = 0, the critical line is obtained R ∼ O ˙ ˙ In Eq. (14) we used = +(1 )∂2, noting that by equating c(0) = h(0), and solving for α. From the L Li −Ti x Li ≡ analytical solutionF weF find that the smallest critical α- ∂x[V 0(x) + i∂x] annihilates its equilibrium distribution p (x, 0), pT (x, 0) = 0. Equation (15) leads us, after value is α = 3, approached for infinitesimal temperature i Li i quenches, 0. integration by parts, to the following integral expression Ti → For small variations of σ away from zero, the topology for ˙i(0): F of the short-time phase diagram remains unchanged. The 2 generic effect of σ > 0 on the critical line is shown by the ˙ (1 i) ∞ i(0) = − T dx pi(x, 0)V 00(x) . (16) dashed lines and black arrows in Fig. 2(a), for values of σ F i T Z−∞ up to unity. We observe that slightly increasing σ moves Evaluating Eqs. (16) and (9) for i = h, c, we obtain (t) the critical line to higher values but does not change the in the short time limit, see Eq. (13). For σ = 0 we solveR phase diagram qualitatively. Eq. (16) explicitly, which gives When σ is decreased to negative values, a more com- plex behaviour emerges, shown by the dotted lines and 2 ( i 1) Γ(1 1/α) white arrows in Fig. 2(a): For initial temperatures close ˙i(0) = (1 α) T − − , (17) F − 2/α Γ(1 + 1/α) to equilibrium 1, the critical line decreases slightly, i i T to α-values belowT ≈ 3. For quenches far from equilibrium, where Γ(x) denotes the gamma function [18]. According on the other hand, there is a considerable shift of the to Eq. (13), whether the hot or the cold system relaxes critical line to higher α. faster at short times is determined by the sign of ˙ (0). We now turn to the analysis of the long-time limit R As a function of α and i it is therefore instructive to t 1 which requires different methods, described in the T draw a “phase diagram”, marking the different regions following. When the spectrum of is discrete, the relax- in parameter space of initially faster heating [ ˙ (0) > 0] ation of the probability densities Lp to p is exponential R i f and initially faster cooling [ ˙ (0) < 0]. in the long-time limit. As a result, the densities p are R i 4 determined by the leading right eigenfunctions of and 4 L (a) 0.2 (b) their corresponding eigenvalues [3]. These quantities are cooling faster
obtained from the non-Hermitian eigenvalue problem α
R 0 rµ = λµrµ , †lµ = λµlµ , (18) crossover L L 3 heating faster −0.2 where lµ and rµ are the left and right eigenfunctions, respectively, and λµ with λ0 = 0 > λ1 > λ2 > . . . are the 0.1 1 5 0 0.1 0.2 0.3 associated eigenvalues. Note that the right eigenfunction T t r0 with eigenvalue λ0 = 0 is given by the steady-state FIG. 3. (a) Superimposed short- and long-time phase dia- distribution r0 = pf and l0 = 1. The eigenfunctions form a complete bi-orthogonal basis with orthonormality grams for σ = 0. The dash-dotted line shows the short-time critical line, the solid line shows the long-time critical line. relations The crossover region is given by the cross-hatched region. The ∞ different coloured dots correspond to the parameter values for lµ rν = dx lµ(x)rν (x) = δµν . (19) the plots in Fig. 3(b). (b) Logarithmic ratio (t) from dif- h | i R Z−∞ ferent numerical methods for h = 3 and α = 3, 3.3 and 3.5, in blue, green and red, respectively.T The dash-dotted lines Expanding p in Eq. (12) in the right eigenbasis of we i show the results from the operator method, the solid lines obtain in the long-time limit t 1, L are results obtained from simulation of the Langevin equa- tion. The black, dashed lines correspond to the short- and p (x, t) p (x) + c eλµtr (x) , (20) i ∼ f i,µ µ long-time asymptotics. where µ is the lowest number for which ci,µ lµ pi(0) = 0. Because our problem is symmetric with≡ respect h | toi the 6 cooling becomes finite and is completely enclosed by the parity operation x x, c vanishes, so that µ = 2, see i,1 critical line [dotted lines in Fig. 2(b)]. The sensitive de- Sec. I in the SM [16]→ for − the case of a harmonic potential. pendence of the relaxation dynamics on negative values All higher order terms in Eq. (20) that play a role at finite of σ, observed both in the short- and long-time limits, times are exponentially suppressed in the long-time limit must be a consequence of the emergence of bi-stability of considered here. Using Eqs. (2) and (20) we find that the potential V (x), Eq. (4). The existence of two poten- (t) approaches a constant for t 1 that depends ∞ tial minima gives rise to multiple relaxation time scales onlyR on the coefficients c : R i,2 associated with the relaxation within the same minimum and across the two minima. ch,2 (t) 2 log . (21) Our short- and long time analyses reveal the exis- R ∼ c ≡ R∞ c,2 tence of distinct critical lines in the short- and long- time limits. This, in turn, results in an overlap between Hence, the relative magnitude of the free-energy differ- ences is determined by the initial overlap between the left the faster-heating and faster-cooling regions at short and eigenvector l2 of and the initial distributions pi(x, 0) long times, giving rise to a crossover region in the phase before the temperatureL quench [3]. diagram. In the crossover region, the hot system initially We determine c by solving the eigenvalue problem relaxes faster [ (t) < 0], but is eventually overtaken by i,2 R (18) numerically, discretising it on an evenly spaced, fi- the initially colder system [ (t) > 0]. Hence, there must R nite lattice with small lattice spacing. Equations (18) be at least one finite time tc > 0 where (tc) = 0, i.e. the R then become matrix eigenvalue problems involving large, system crosses over from faster cooling to faster heating. non-symmetric matrices, whose left and right eigenvec- Figure 3(a) shows the superimposed short- and long- tors are approximations of the left and right eigenfunc- time phase diagrams for σ = 0 featuring the crossover tions rµ and lµ. region (cross-hatched area). The dash-dotted and solid Figure 2(b) shows the long-time phase diagram for lines show to the critical lines from Figs. 2(a) and (b), σ = 0 obtained from numerically computing ci,2 and eval- respectively. uating in Eq. (21). The general structure of the long- In order to study the behaviour of (t) in the crossover time phaseR∞ diagram is qualitatively similar to that of the region, and to validate our previousR results, we perform short-time phase diagram in Fig. 2(a), featuring regions a numerical analysis of the finite-time evolution of (t). of faster heating [ > 0] and faster cooling [ < 0]. We focus on a few points in the phase diagram, shownR as For long times, however,R∞ the critical line [solidR∞ line in the differently coloured dots in Fig. 3(a), where we expect Fig. 2(b)], where the behaviour switches from faster heat- qualitatively different behaviours. More specifically, for ing to faster cooling, is shifted to slightly higher values. the parameter sets represented by the blue and red dots, We observe that increasing σ does not alter the phase di- we expect heating and cooling, respectively, to be faster, agram qualitatively, but merely pushes the critical line to both for short and for long times. By contrast, for the larger α-values [dashed lines in Fig. 2(b)]. Negative σ, on parameters of the green dot we expect at least one finite- the other hand, leads to a qualitative change in the phase time crossover from faster cooling to faster heating. diagram: For σ < 0, the region of asymptotically faster For the finite-time analysis we use two different nu- 5 merical methods. First, we obtain an approximation of rium where heating is initially faster but asymptotically pi(x, t) by using the discretised analogue of Eq. (12) ob- slower than cooling. tained with the discretisation scheme discussed earlier. In conclusion, against common belief [13–15], - The second method approximates the probability den- equidistant thermal relaxation for overdamped diffusionsF sity pi(x, t) by means of a Langevin approach [19]: We in anharmonic, single-well potentials V (x) allows for both simulate a large number of trajectories xi(t), i = h, c, faster heating and faster cooling. To capture this be- following the dynamicsx ˙ (t) = V 0(x) + ξ(t), where ξ(t) i − haviour, we obtained the short- and long-time phase di- is a Gaussian white-noise signal with correlation func- agrams [Figs. 2(a) and (b)]. The phase diagrams de- tion ξ(t)ξ(t0) = 2δ(t t0). The initial values x (0) are h i − i termine where in the ( , α)-parameter space heating or sampled from the equilibrium distributions pi(x, 0) prior cooling are faster. WeT showed that the short- and long- to the temperature quench. The Langevin equation is time phase diagrams are different, and that the faster- solved numerically using an Euler-Maruyama scheme [20] heating and faster-cooling regions overlap. Superimpos- with a small time step. The probability densities pi(x, t) ing the two, we localised a crossover region [Fig. 3(a)] are then computed by generating histograms over all lo- where cooling is initially faster but the rate of heating cations xi(t) at discrete times t. eventually overtakes. Outside of the crossover region, Both these methods yield pi(x, t) from which we then we found no crossings, suggesting that the short- and calculate (t). The virtue of using two different methods R long-time asymptotics faithfully characterise the relative is to make sure that our results are not affected by the relaxation speeds. The critical lines separating the pa- discretisation of the time and spatial domains. rameter regions with different behaviours are only weakly Figure 3(b) shows the so-obtained (t), where the R perturbed by an additional harmonic term in the poten- colours of the curves correspond to the colours of the dots tial V (x), as long as the latter remains single-well, i.e. in Fig. 3(a). The dash-dotted lines show (t) calculated σ > 0. from the discretised operator . The lighter,R solid lines It would be interesting to test the relaxation-speed show the corresponding resultsL from the Langevin ap- crossover in state-of-the-art experiments [7, 21] and thus proach. Also shown are the short- and long-time asymp- to reproduce our phase diagram under experimental con- totes (dashed lines). We observe that the asymptotes ditions. On the theoretical side, it would be desirable to represent a good characterisation of the dynamics of (t) understand the dynamical origins of the different relax- for all times. In particular, there are no finite-time cross-R ation behaviours [22]. This might lead to optimisation ings (t ) = 0 for the parameter values outside of the c methods for the potential to achieve faster heating or crossoverR region in Fig. 3(a), i.e. for the blue and red cooling, perhaps in the spirit of first-passage time opti- curves. Inside the crossover region [see green curve in misation [23, 24]. Fig. 3(b)] we observe only a single crossing. Furthermore, there is good agreement between the results from the different numerical methods and the asymptotic results. Note that the deviations between the ACKNOWLEDGMENTS equally-coloured curves become larger for longer times. The reason is that for long times, the individual free- energy differences i(t) in Eq. (2) become exponentially We thank John Bechhoefer and Massimiliano Espos- small, so that theF relative errors increase as t becomes ito for discussions. Funding from the European Re- large. search Council within the project NanoThermo (ERC- Finally, we note that far from equilibrium, for 2015-CoG Agreement No. 681456) and the Foundational Tc ≈ 0.0229 and h 5.50, the short- and long-time critical Questions Institute within the project Information as a lines cross [notT ≈ shown in Fig. 3(a)]. As a consequence, fuel in colloids and superconducting quantum circuits there is an inverted crossover region very far from equilib- (FQXi-IAF19-05) is gratefully acknowledged.
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[13] A. Lapolla and A. Godec, Physical Review Letters 125, [19] N. G. Van Kampen, Stochastic processes in physics and 110602 (2020). chemistry, Vol. 1 (Elsevier, 1992). [14] S. K. Manikandan, arXiv preprint arXiv:2102.06161 [20] P. E. Kloeden and E. Platen, Numerical Solution of (2021). Stochastic Differential Equations (Springer, 1992). [15] T. Van Vu and Y. Hasegawa, arXiv preprint [21] S. Ciliberto, Physical Review X 7, 021051 (2017). arXiv:2102.07429 (2021). [22] In Sec. III of [16], we trace back the long-time relaxation [16] See Supplemental Material [url] for details, which in- asymmetry in harmonic potentials to the convexity of the cludes the additional Ref. [25]. inverse of the distance function . [17] H. Risken, in The Fokker-Planck Equation (Springer, [23] V. V. Palyulin and R. Metzler, JournalF of Statistical Me- 1996) pp. 63–95. chanics: Theory and Experiment 2012, L03001 (2012). [18] DLMF, NIST Digital Library of Mathematical Func- [24] M. Chupeau, J. Gladrow, A. Chepelianskii, U. F. Keyser, tions, http://dlmf.nist.gov/, Release 1.1.2 of 2021-06- and E. Trizac, Proceedings of the National Academy of 15, f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, Sciences 117, 1383 (2020). B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, [25] U. M. Titulaer, Physica A: Statistical Mechanics and its B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. Applications 91, 321 (1978). Supplemental Material for: Relaxation-speed crossover in anharmonic potentials
Jan Meibohm,1 Danilo Forastiere,1 Tunrayo Adeleke-Larodo,1 and Karel Proesmans1, 2 1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg 2Hasselt University, B-3590 Diepenbeek, Belgium
In this Supplemental Material, we outline a computation of the c2-coefficient for over- and un- derdamped Langevin systems in harmonic potentials. Furthermore, we discuss how the relaxation asymmetry after a -equidistant temperature quench relates to the Markovian Mpemba effect de- scribed in the literature.F Finally, we show that in the long-time limit and for harmonic potentials, the heating-cooling asymmetry can be traced back to a convexity property of the inverse of the distance measure . F
I. ANALYTICAL EXPRESSION OF c2 FOR II. CONNECTION WITH MARKOVIAN HARMONIC POTENTIALS MPEMBA EFFECT
In this section, we outline the derivation of the c2- coefficient for over- and underdamped diffusion in har- (a) (b) monic potentials. In particular, we show that the c2-
coefficients are linear in the temperature ratio . To | 2
T c calculate c2, we need to find the left eigenvectors associ- | ated with the Fokker-Planck operator [Eq. (5) in the main text]. L 0 We start by discussing the overdamped limit. The po- 0 1 Tc Th ←Fc 0 Fh→ tential V (x) [Eq. (4) in the main text] is chosen to be har- T F monic V (x) = x2, i.e., σ = 0 and α = 2. For harmonic potentials, the Fokker-Planck equation can be solved an- Figure 1. The Markovian Mpemba effect and the long-time alytically [1]. The first three left eigenfunctions read limit of the -equidistant temperature quench. (a) Schematic of the magnitudeF of c as a function of the initial temperature 2 2 l0(x) = 1 , l1(x) = x , l2(x) = 2x 1 . (S1) . The blue and red solid lines show a hypothetical c2( ). − T T The associated eigenvalues are λ = 0, λ = 2 and λ = The circles correspond to a temperature pair that exhibits the 0 1 2 Markovian Mpemba effect, the squares show an -equidistant 4, respectively. With the initial distributions− given by temperature pair [see Fig. 1(b)]. The dashed linesF showcase p−( ) = exp( x2/ )/√ π, we obtain from T − T T a linear c2( ). (b) Same schematic as in Fig. 1(a), but as a function ofT the free-energy difference . The squares repre- cµ( ) = lµ p( ) (S2) T h | T i sent a pair of -equidistant temperatures.F the coefficients F
c0( ) = 1 , c1( ) = 0 , c2( ) = 1 . (S3) In this section, we discuss the relation between the T T T T − We observe that c is linear in the dimensionless temper- relaxation asymmetry in -equidistant quenches and the 2 Markovian Mpemba effect.F ature ratio = Tc/h/Tf for overdamped diffusion in a harmonic potential.T Clearly, the relaxation asymmetry and the Markovian The calculation for the underdamped case goes along Mpemba effect share similarities. After all, both effects similar lines. For the computation of the left eigenfunc- refer to the thermal relaxation of two identical systems tions of the Kramers equation we refer to, e.g., Appendix at different initial temperatures towards the same final A of Ref. [2]. From Eq. (S2) one finds temperature. In particular, this allows to use similar methods for their description: In the main text, we used c0( ) = 1 , c1( ) = 0 , c2( ) = N2 ( 1) , (S4) the method of calculating the coefficients c to deter- T T T T − 2 with mine the long-time behaviour of the cold and hot sys- 2 tems after an -equidistant temperature quench. This 2 F N2 = 1 + 4 ζ ζ 1 . (S5) approach for obtaining the asymptotic rates of heating arXiv:2107.07894v1 [cond-mat.stat-mech] 16 Jul 2021 − − and cooling after temperature quenches has become pop- Here, ζ = mγ2/(8k) is the dimensionlessp damping ra- ular through its application to the Markovian Mpemba tio with particle mass m, friction coefficient γ and har- effect in Ref. [3]. monic couplingp k. Equations (S4) show that also in the Apart from the similarities, there are two important underdamped limit, the coefficient c2 is linear in the tem- differences that we discuss in the following. perature ratio . First, the perhaps most obvious difference between the Note that, uponT taking the overdamped limit ζ , two effects is the ordering of the temperature ratios. For → ∞ the second term in Eq. (S5) vanishes and we recover the -equidistant quench we have c < 1 < h. For the Eqs. (S3). MarkovianF Mpemba effect, on theT other hand,T one has 2
1 < c < h, or c < h < 1 for the inverse Mpemba then c 1 = h 1 [see squares in Fig. 1(a)]. In other effectT [3]. ItT was shownT T in Ref. [3] that a sufficient con- words,|T −-equidistance| 6 |T − | does not imply -equidistance, F T dition to observe the Mpemba effect is that c2( ) is a nor equidistance with respect to any other generic dis- non-monotonic function of the initial temperatureT ratio tance measure. Note that this argument also holds for a > 1. In this case, there always exist temperature pairs generic, nonlinear c2( ). 1T < < so that c ( ) > c ( ) . As a result, In summary, althoughT the setup for the two effects is Tc Th | 2 Tc | | 2 Th | the initially hotter system (with initial temperature h) similar, there are two important differences: The tem- relaxes asymptotically faster than the colder one (withT perature ordering and the dependence on the distance initial temperature c). The condition c(t) < h(t) for measure. all times, as discussedT in the main text andF in Refs.F [4, 5], is somewhat stronger, as it is supposed to hold also at fi- nite times. For the sake of comparison with the Mpemba III. EFFECTS OF DIFFERENT DISTANCE effect, we here restrict ourselves to the long-time limit, MEASURES IN THE LONG-TIME LIMIT. where the c2-coefficient is a faithful measure for both phenomena. In this section, we show that in the long-time limit and In Fig. 1(a), c2 is shown schematically as a function for harmonic potentials, the asymmetry between heating of . Here, the| Markovian| Mpemba effect corresponds and cooling in -equidistant temperature quences can be to theT non-monotonicity of the red branch: We may find traced back toF the convexity of the inverse of . F pairs of initial temperatures c and h (white circles) To this end, we consider a generic measure f( ) for the T T T so that c2( c) > c2( h) . The equivalent phenomenon, distance from equilibrium, assuming that f has a single but for |
[1] H. Risken, in The Fokker-Planck Equation (Springer, [4] A. Lapolla and A. Godec, Physical Review Letters 125, 1996) pp. 63–95. 110602 (2020). [2] U. M. Titulaer, Physica A: Statistical Mechanics and its [5] S. K. Manikandan, arXiv preprint arXiv:2102.06161 Applications 91, 321 (1978). (2021). [3] Z. Lu and O. Raz, Proceedings of the National Academy of Sciences 114, 5083 (2017).