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 cooling.R 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 .