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ORIGINAL RESEARCH article

Front. Phys., 10 October 2022
Sec. Soft Matter Physics
This article is part of the Research Topic Active and Externally Driven Granular Matter View all 8 articles

Mpemba-like effect protocol for granular gases of inelastic and rough hard disks

  • 1Departamento de Física, Universidad de Extremadura, Badajoz, Spain
  • 2Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, Badajoz, Spain

We study the conditions under which a Mpemba-like effect emerges in granular gases of inelastic and rough hard disks driven by a class of thermostats characterized by the splitting of the noise intensity into translational and rotational counterparts. Thus, granular particles are affected by a stochastic force and a stochastic torque, which inject translational and rotational energy, respectively. We realize that a certain choice of a thermostat of this class can be characterized just by the total intensity and the fraction of noise transferred to the rotational degree of freedom with respect to the translational ones. Firstly, Mpemba effect is characterized by the appearance of a crossing between the temperature curves of the considered samples. Later, an overshoot of the temperature evolution with respect to the steady-state value is observed and the mechanism of Mpemba effect generation is changed. The choice of parameters allows us to design plausible protocols based on these thermostats for generating the initial states to observe the Mpemba-like effect in experiments. In order to obtain explicit results, we use a well-founded Maxwellian approximation for the evolution dynamics and the steady-state quantities. Finally, theoretical results are compared with direct simulation Monte Carlo and molecular dynamics results, and a very good agreement is found.

1 Introduction

Since the Antiquity, the fact that water could start freezing earlier for initially hotter samples was observed and commented by very influential people of different epochs like Aristotle [1], Francis Bacon [2], or René Descartes [3]. This counterintuitive phenomenon contradicts Isaac Newton’s formulation of its well-known cooling’s law [4, 5], but otherwise it is part of the popular belief in cold countries. The scientific community started to pay attention to this effect since the late 60s of the last century thanks to its accidental rediscovery by a Tanzanian high-school student, Erasto B. Mpemba. Later, he and Dr. Denis Osborne reported their findings [6, 7] and since then the effect is usually known as Mpemba effect (ME).

Whereas the original tested system for ME has been water [637], it is still under discussion and no consensus about its occurrence has been agreed [3840]. In fact, the statistical physics community is currently paying attention to Mpemba-like effects that have been described in a huge variety of complex systems in the last decades, such as ideal gases [41], molecular gases [4244], gas mixtures [45], granular gases [4652], inertial suspensions [53, 54], spin glasses [55], Ising models [5658], non-Markovian mean-field systems [59, 60], carbon nanotube resonators [61], clathrate hydrates [62], active systems [63], or quantum systems [64]. The theoretical approach to the fundamentals of the problem has been done via different routes like Markovian statistics [6569] or Landau’s theory of phase transitions [70]. Recently, in the context of a molecular gas under a nonlinear drag force, new interpretations and definitions of ME from thermal and entropic point of views, as well as a classification of the whole possible phenomenology, have been carried out [44]. In addition, ME has been experimentally observed in colloids [71, 72], proving that it is a real effect present in nature.

The very first time that ME was observed theoretically in granular gaseous rapid flows was in [46]. The considered system was a set of inelastic and smooth hard spheres (with constant coefficient of normal restitution) heated by a stochastic thermostat, the effect arising by initially preparing the system in far from Maxwellian states. The same type of initial preparation was applied to the case of molecular gases with nonlinear drag [42, 44]. Essentially, the temperature evolution depends on the whole moment hierarchy of the velocity distribution function (particularly on the excess kurtosis, or fourth cumulant, and, more weakly, on the sixth cumulant), this dependence giving rise to the possible appearance of ME.

On the other hand, there is no need to consider an initial velocity distribution function (VDF) far from the Maxwell–Boltzmann one if the temperature is coupled to other basic variables that can be fine-tuned in the initial preparation of the system. This occurs in the case of a monocomponent granular gas made by inelastic and rough hard spheres thermostatted by a stochastic force [47], as well as in driven binary mixtures of either molecular or inelastic gases [45, 48]. In those systems, one does not need to invoke strong nonGaussianities, since the temperature relaxation essentially depends on the rotational-to-translational temperature ratio (in the case of rough particles) or on the partial component temperatures (in the case of mixtures). A similar situation applies in the presence of anisotropy in either the injection of energy [51] or in the velocity flow [53]. However, there is still a lack of protocol defining a possible nearly realistic preparation of the initial states for a granular or molecular gas in homogeneous and isotropic states. Unlike other memory effects, such as Kovac’s effect [73, 74], ME has not a predefined way to elaborate a protocol.

In this work, we have addressed the latter preparation problem for a specific model of granular gases. We consider a monodisperse granular gas of inelastic and rough hard disks, where inelasticity is parameterized via a constant coefficient of normal restitution, α, and the roughness is accounted for by a coefficient of tangential restitution, β, assumed to be constant as well. Disks are “heated” by a stochastic thermostat which injects energy to both translational and rotational degrees of freedom through a combination of a stochastic force and a stochastic torque, both with properties of a white noise. The relative amount of energy injected to the rotational degree of freedom, relative to that injected to the translational degrees of freedom, can be freely chosen. Therefore, we will denote this thermostat as splitting thermostat (ST). The quantity coupled to the temperature that will monitor the possible occurrence of ME will be the rotational-to-translational temperature ratio, as in [47], where, however, a stochastic torque was absent. This double energy-injection based on ST allows us to fix the initial conditions of the variables that play a role in the evolution process, namely the temperature and its coupling. A side effect of providing energy to the rotational degree of freedom is that it favors the possibility of an overshoot of the temperature with respect to its steady-state value. This might cause ME, even in the absence of a crossing between the temperatures of the two samples [44]. Therefore, the protocol must be adapted to this specific phenomenon.

It is worth saying that our theoretical approach is based on a Maxwellian approximation (MA), that is, we assume that both transient and steady-state VDFs are close to a two-temperature Maxwellian. This approach is founded on previous works for the case of zero stochastic torque [75] and on preliminary results for the system at hand [76]. Moreover, the two-dimensional characterization of the physical system is thought to be plausible for hopefully being reproduced in some experimental setup. As will be seen, the reliability of our theoretical approach is confirmed by computer simulations via the direct simulation Monte Carlo (DSMC) method and event-driven molecular dynamics (EDMD).

The paper is structured as follows. In Section 2, the model system for a granular gas of inelastic and rough hard disks thermostatted by stochastic force and torque is introduced. Also, explicit evolution equations and expressions for the steady-state dynamic variables are shown under the MA, and the theoretical results are compared with DSMC and EDMD. Section 3 collects the definition and necessary conditions for ME to occur taking into account the emergence or not of overshoot during evolution. Subsequently, and based on the analysis of this section, two different protocols are presented for observing ME in cases without and with overshoot, respectively. This discussion is accompanied by its proper comparison with simulation results. Finally, concluding remarks are presented in Section 4.

2 The model

We consider a set of mechanically identical inelastic and rough hard disks of mass m, diameter σ, and reduced moment of inertia κ ≡ 4I/2 (I being the moment of inertia). The translational velocities lie on the xy plane, i.e., v=vxx̂+vyŷ, while the angular velocities point along the orthogonal z axis, ω=ωẑ. Inelasticity and roughness are characterized by the coefficients of normal and tangential restitution, α and β, respectively, which are assumed to be constant and defined as [77].

σ̂v12=ασ̂v12,σ̂v12=βσ̂v12,(1)

Where σ̂(r2r1)/|r2r1| is the unit intercenter vector along the line of centers from particle 1 to particle 2, σ̂σ̂×ẑ is orthogonal to σ̂, v12v1v2 is the relative velocity between particles 1 and 2, and primed quantities account for their postcollisional values. Because of their definitions, the ranges of the coefficients of restitution are 0 ≤ α ≤ 1 and −1 ≤ β ≤ 1, α = 1 corresponding to elastic collisions, β = −1 describing a perfectly smooth disks, and β = 1 standing for completely rough disks. In fact, total kinetic energy is only conserved if α = |β| = 1 [7780].

In the case α ≠ 1 or |β| ≠ 1, that is, when kinetic energy is dissipated upon collisions, the undriven system will evolve up to a completely frozen state. In order to avoid that quench, we will force the particles to externally receive energy via a homogeneous stochastic force Fwn and a homogeneous stochastic torque τwn that inject translational and rotational kinetic energy, respectively, with the properties of a white noise. That is,

Fiwnt=0,FiwntFjwnt=Im2χtr2δijδtt,(2a)
τiwnt=0,τiwntτjwnt=mIχrot2δijδtt,(2b)

where I is the 2 × 2 identity matrix, i and j are particle indices, and χtr2 and χrot2 are the intensities of the noises applied to the translational and rotational degrees of freedom, respectively. The combination of the stochastic force and torque is characterized by the pair of parameters (χtr2,χrot2) and defines the ST, as described in Section 1. In Section 2.1 we will introduce a more manageable pair of equivalent parameters. An illustration of the system is represented in Figure 1.

FIGURE 1
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FIGURE 1. (A) Illustration of the system, where the green horizontal straight arrows and the blue curved ones represent the stochastic force Fwn and the stochastic torque τwn in Eqs. 2, respectively. (B) Mechanism of injection-dissipation of energy in the introduced model. Dissipative collisions produce a cooling effect on the translational and rotational temperatures (Ttr and Trot), together with a transfer between the translational and rotational energies (equipartition effect). Additionally, the external white-noise force and torque inject energy (heating effect).

To dynamically describe the system, we will work under the assumptions of the homogeneous Boltzmann–Fokker–Planck equation (BFPE). That is, we consider a homogeneous and isotropic gas in a dilute regime, such that the evolution due to the collisional process is determined by just binary collisions, assuming Stosszahlansatz (or molecular chaos). The BFPE for this collisional model together, with the ST, is written as follows

tfv,ω;tχtr22v2fv,ω;tχrot22ω2fv,ω;t=Jv,ω|f,(3)

where f is the one-particle VDF, J[v, ω|f] is the usual Boltzmann collision operator for hard disks defined as [7880].

Jv1,ω1|f=nσdv2dω2+dσ̂|v12σ̂|×fv1,ω1;tfv2,ω2;tα|β|fv1,ω1;tfv2,ω2;t,(4)

where n is the number density, the subscript + in the integral over σ̂ means the constraint v12σ̂>0, and double primed quantities are precollisional velocities, which are given by [7880].

v1/2=v1/21+α12v12σ̂σ̂+κ1+κ1+β12v12σ̂σ2ω1+ω2σ̂.(5)

2.1 Dynamics

The time evolution of the system is fully described by the BFPE, Eq. 3, which allows one to determine the dynamics of macroscopic quantities. The most important and basic quantities to study the dynamics of the system will be the translational and rotational granular temperatures defined at a certain time t as

Ttrt=12mv2,Trott=Iω2,(6)

where the notation ⟨⋅⟩ means the average over the instantaneous VDF,

Xv,ω;t=n1dvdωXv,ω;tfv,ω;t.(7)

One should notice that the partial noise intensities χtr2 and χrot2 affect directly the evolution equations of Ttr(t) and Trot(t), respectively. On the other hand, those quantities are coupled due to the transfer of energy during collisions (see Figure 1B). The description of the dynamics from Ttr(t) and Trot(t) is equivalent to consider the mean granular temperature, T(t), and the rotational-to-translational temperature, θ(t), defined as follows

Tt=23Ttrt+13Trott,θt=TrottTtrt.(8)

In the definition of T(t) we have taken into account that there are two translational and one rotational degrees of freedom. This new pair of variables will be useful to study ME, which will be related to the evolution of the mean granular temperature T(t). In addition, this change of dynamical quantities in Eq. 8 induces a change of parameters describing the ST. Thus, we introduce the total noise intensity, χ2, and the rotational-to-total noise intensity ratio, ɛ, as

χ2=χtr2+I2mχrot2,ε=I2mχrot2χ2.(9)

Notice that χtr2=(1ε)χ2. Therefore, 0 ≤ ɛ ≤ 1, ɛ = 0 and ɛ = 1 corresponding to the purely translational and the purely rotational thermostat, respectively. The total noise intensity χ2 is unbounded from above and, by dimensional analysis, can be equivalently characterized by a noise temperature

Twnmχ2πnσ2/3.(10)

Therefore, from now on the ST will be characterized by the pair (Twn, ɛ). In terms of the new parameters, the BFPE, Eq. 3, reads

tfv,ω;tνwnTwn4m1εv2+ε2mIω2fv,ω;t=Jv,ω|f,(11)
νwn=2nσπTwnm(12)

being a reference noise-induced collisional frequency.

Inserting the definitions of partial granular temperatures, Eq. 6, into Eq. 11, one obtains

tTtr=ξtrTtr+1ε2νwnTwn,tTrot=ξrotTrot+ενwnTwn,(13)

where

ξtr=m2nTtrdvdωv2Jv,ω|f,(14a)
ξrot=InTrotdvdωω2Jv,ω|f(14b)

are the translational and rotational energy production rates [78, 80].

In terms of the quantities defined in Eq. 8, Eqs. 13 become

tT̃=ζT̃+13νwn,(15a)
tθ=θξtrξrotνwn2+θ6T̃θε2+θ,(15b)

where T̃T/Twn and

ζ23ξtrTtrT+13ξrotTrotT=2ξtr+ξrotθ2+θ(16)

is the cooling rate.

According to Eqs. 15, the steady-state quantities T̃st=Tst/Twn and θst=Trotst/Ttrst satisfy the conditions

ζstT̃st=13νwn,εξtrst=1ε2ξrotstθst,(17)

which imply a balance between collisional cooling and external heating. The steady-state temperature can be used to define a reduced temperature TT̃/T̃st=T/Tst and a reduced time t=12νstt, where

νst=2nσπTtrstm=νwn3T̃st2+θst(18)

is the steady-state collision frequency. More in general, the time-dependent collision frequency is

νt=2nσπTtrtm=νwn3T̃t2+θt=νstGTt,θt,(19a)
GT,θT2+θst2+θ.(19b)

The above collision frequency can be used to nondimensionalize the energy production rates as

μ20ξtrν,μ02ξrot2ν,(20)

where

μk=dcdwckwJc,w|f(21)

are the reduced collisional moments. In Eq. 21, c=v/2Ttr/m and w=ω/2Troot/I are the reduced translational and angular velocities, respectively, and J(2Ttr/m)2Troot/I/(nν)J is the reduced collision operator. Thus, the steady-state conditions (17) become

2μ20st+μ02stθst=2+θst3T̃st3/2γst,εμ20st=1εμ02stθst.(22)

Using these dimensionless definitions, Eqs. 15 yield

12tlnT=2GT,θμ20+μ02θ2+θ+2Tμ20st+μ02stθst2+θst,(23a)
12tlnθ=GT,θμ202μ021T2+θ2+θstθstθθε2+θθstε2+θstμ20st2μ02st,(23b)

where use has been made of Eqs. 17, 22.

According to the definition of collisional moments, Eq. 21, they depend on the whole VDF. This implies that Eqs. 23 do not make a closed set of equations. The same applies to the steady-state solution, Eqs. 22. This shortcoming, however, can be circumvented if an approximate closure is applied. This is the subject of section 2.2.

2.2 Maxwellian approximation

In order to get explicit results from Eqs. 22, 23 by using the simplest possible closure, we resort to the two-temperature MA

fv,ωnm2πTtrI2πTrotexpmv22TtrIv22Trot.(24)

This approximation does a very good job in the three-dimensional case with ɛ = 0 [75] and it is reasonably expected to perform also well in the case of disks with ɛ ≠ 0.

Within this approximation, the relevant collisional moments can be evaluated with the result [7880].

μ20=1α22+K2κ1β21+θκ+1θ,(25a)
μ02=K21β2κ1+κθ+1θ1,Kκ1+β1+κ2.(25b)

Solving Eqs. 22, we get the steady-state expressions

θst=κ2K1α2ε+K1+κ1β1ε1+κ+2κ1,T̃st=2+θst3γst2/3,(26)

with

γst=1α2+K1+κ2κ1βκ+θst.(27)

In particular,

ε=0θst=1+β2+κ11β,(28a)
ε=1θst=21+K11α2+κ21β1+β.(28b)
Eq. 28a agrees with a previous result [78]. Notice that, for the special value ɛ = 0, θst is independent of the coefficient of normal restitution α both for disks and spheres [47, 75, 7880]. However, this property is broken down when energy is injected into the rotational degree of freedom (ɛ ≠ 0).

From Eqs. 26 one can observe that θst is independent of Twn and, at given α and β, it is a monotonically increasing function of ɛ. This is physically expected since, by growing ɛ, we are increasing the relative amount of rotational energy injected with respect to the total energy; therefore, it is presumed that the stationary value of Trot rises with respect to Ttr at fixed Twn. Thus, the most disparate values of θst correspond to ɛ = 1 and 0, their difference being plotted in Figure 2 as a function of α and β.

FIGURE 2
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FIGURE 2. Difference θst(ɛ = 1) − θst(ɛ = 0) [Eq. 28] in the plane (α, β).

It is interesting to note that, in the MA, Eq. 24, one simply has.

2μ20+μ02θ=γst+K1β1+κ2κθθst,(29a)
μ202μ02=1ε2+θstθstγst2K1+β41+2θθstθθst.(29b)

As a consequence, Eqs. 23 can be recast as

12tlnT=ΦT,θγstGT,θ2+θ1/T2+θstGT,θK1β1+κ2κθθst2+θ,(30a)
12tlnθ=1ε2+θstθstγst2GT,θ1T2+θ2+θstθstθθε2+θθstε2+θstGT,θK1+β41+2θθstθθst.(30b)
Eqs. 30 make a closed set of two ordinary differential equations that can be (numerically) solved with arbitrary initial conditions T(0)T0 and θ(0) ≡θ0. Although the theoretical results have been derived for arbitrary values of the reduced moment of inertia κ, henceforth all the graphs are obtained for the conventional case of a uniform mass distribution of the disks, i.e., κ=12.

Figures 3, 4 show density plots of Φ(T, θ) for α = 0.9 and 0.7, respectively. In each case, two values of β (0 and −0.7) and three values of ɛ (0, 0.5, and 1) are considered. We observe that, typically, Φ(T, θ) increases with increasing θ at fixed T, while it decreases with increasing T at fixed θ.

FIGURE 3
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FIGURE 3. Density plots of Φ(T*, θ) for α = 0.9 and (A) β = 0, ɛ = 0; (B) β = 0, ɛ = 0.5; (C) β = 0, ɛ = 1; (D) β = −0.7, ɛ = 0; (E) β = −0.7, ɛ = 0.5; and (F) β = −0.7, ɛ = 1. The contour lines (solid for Φ < 0, dashed for Φ > 0) are separated by an amount ΔΦ = 0.1. The thick solid line is the locus Φ(T*, θ) = 0. It intercepts the (brown dash-dotted) vertical line T* = 1 at θ = θst.

FIGURE 4
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FIGURE 4. Same as Figure 3 but for α = 0.7.

2.3 Comparison with simulation results

In order to check the validity of the MA, we have compared our theoretical predictions against DSMC and EDMD simulation results both for transient and steady-state values.

The DSMC algorithm used is based on the one presented in, e.g., [81, 82], and adapted for the model presented in this work. For our DSMC simulations we have dealt with N = 104 particles and chosen a time step Δt = 4 × 10–5/νwn. In addition, the way of implementing the stochastic force and torque in the EDMD code is based on the approximate Green function algorithm [83], as applied to the ST. We have chosen N = 3.6 × 103 particles, a density 2 = 5 × 10–4 (implying a box length of L/σ ≈ 1897.37), and a time step Δt ≈ 4 × 10–4/νwn. No instabilities were observed.

In Figure 5, results for the steady-state values T̃st and θst as functions of ɛ and different values of α and β are presented. Simulation results for DSMC and EDMD come from averages over 100 replicas and over 50 data points, once the steady state is ensured to be reached. A very good agreement between DSMC and EDMD with expressions in Eqs. 26 is observed. Whereas θst is an increasing function of ɛ, this is not the case, in general, with T̃st, as can be observed in Figure 5E.

FIGURE 5
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FIGURE 5. Steady-state values of the mean temperature T̃st and the rotational-to-translational temperature ratio θst as functions of ɛ. The values of the coefficients of restitution are (A,B): (α, β) = (0.7, 0); (C,D): (α, β) = (0.7, − 0.7); (E,F): (α, β) = (0.9, 0); and (G,H): (α, β) = (0.9, − 0.7). Thick black lines correspond to the theoretical prediction in Eqs. 26, and symbols refer to DSMC (o) and EDMD (×) simulation results for ɛ = 0, 0.1, 0.25, 0.5, 0.8, 0.9, and 1.

As a test of the transient stage, we present in Figure 6 the evolution of T and θ (starting from a Gaussian-generated VDF with T0=1.5 and θ0 = 1) for the same choices of α and β as in Figure 5 and for the representative values ɛ = 0 (translational noise only), 0.8 (both translational and rotational noise), and ɛ = 1 (rotational noise only). We observe again an excellent agreement of the MA, Eqs. 30, with simulation results.

FIGURE 6
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FIGURE 6. Time evolution of the reduced mean temperature T* and the rotational-to-translational temperature ratio θ for ɛ = 0, 0.8, and 1. The initial conditions are T0*=1.5 and θ0 = 1 in all the cases. The values of the coefficients of restitution are (A,B): (α, β) = (0.7, 0); (C,D): (α, β) = (0.7, − 0.7); (E,F): (α, β) = (0.9, 0); and (G,H): (α, β) = (0.9, − 0.7). Lines correspond to the theoretical prediction from Eq. 30, and symbols refer to DSMC and EDMD simulation results.

2.4 Temperature overshoot

As illustrated in Figure 6, the evolution of T(t) for certain initial states might experiment an overshoot T(tO)=1 at a finite time tO, followed by a minimum, and then relax to the steady state from below. This overshoot effect becomes more pronounced as Φ(T0,θ0) takes more negative values, i.e., as ɛ increases and/or θ0 decreases.

In general, at a given initial condition (T0,θ0), there exists a critical value εcr(T0,θ0), such that the T(t) exhibits overshoot if ε>εcr(T0,θ0). The determination of εcr(T0,θ0) within the MA requires the numerical solution of Eqs. 30. We have analyzed those numerical solutions up to t = 15 since the overshoot typically takes place in the first stage of the evolution. Thus, the numerical value ɛ = ɛcr corresponds to tO=15.

Figure 7 shows εcr(T0,θ0) as a function of T0 at a specific value of θ0, namely the one given by Eq. 28a and for the same four pairs of coefficients of restitution as in Figures 36. In each case, the curve εcr(T0,θ0) splits the plane ɛ vs. T0 into two regions: the region above the curve, where the overshoot effect is present, and the one below the curve, where temperature relaxes to the steady-state value from above. Note that the shape of the curve ɛcr in Figure 7 associated with the pair (α, β) = (0.7, −0.7) differs in curvature from the curves associated with the other three pairs.

FIGURE 7
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FIGURE 7. Phase diagram for the emergence of overshoot. The numerical critical value εcr(T0*,θ0) is presented as a function of T0* for θ0 given by Eq. 28a and four different pairs of the coefficients of restitution: (α, β) = (0.7, 0), (0.7, − 0.7), (0.9, 0), and (0.9, − 0.7).

3 Mpemba effect

As already said in Section 1, ME refers to the counterintuitive phenomenon according to which an initially hotter sample of a given fluid relaxes earlier to the steady state than an initially colder one. In a recent paper [44], we distinguished between the thermal ME—where the relaxation process is described by the temperature of the system (second moment of the VDF)— and the entropic ME—where the deviation from the final steady state is monitored by the Kullback–Leibler divergence (thus involving the full VDF). Whereas this distinction is interesting and the relationship between the thermal ME and the entropic ME is not always biunivocal [44], we focus this paper on the thermal version due to its simpler characterization and its relationship with the original results [6]. Morover, only cooling processes will be considered throughout this work.

Let us assume two samples—denoted by A and B—of the same gas, subject to the same noise temperature Trefwn and the same splitting parameter ɛref, so that the final steady-state values T̃refst and θrefst will also be the same. Both samples differ in the initial conditions (T0A,θ0A) and (T0B,θ0B), respectively, where T0A>T0B>1, that is, A refers to the initially hotter sample and we are considering a cooling experiment.

3.1 Standard Mpemba effect

Let us first consider the standard form of thermal ME [42, 4447], where both TA(t) and TB(t) cross over at a certain crossing time tc and then relax from above, i.e., TA(tc)=TB(tc) and TB(t)>TA(t)>1 for t>tc. The emergence of ME can be subdued to the appearance of a crossing time (or an odd number of them) in the absence of any overshoot effect in the thermal evolution [44]. We will refer to this situation as the standard ME (SME). According to the discussion in Section 2.4, the SME implies that εref<min{εcr(T0A,θ0A),εcr(T0B,θ0B)}. The case when a temperature overshoot takes place will be discussed in Section 3.2.

It can be reasonably expected that a necessary condition for the occurrence of the SME is that the initial slope is smaller in sample A than in sample B, i.e.,

ΦT0A,θ0A<ΦT0B,θ0B.(31)

Note here that the usual situation is that both slopes are negative, in which case |Φ(T0A,θ0A)|>|Φ(T0B,θ0B)|. Of course, Eq. 31 is not sufficient for the SME since the latter also depends on how close T0A and T0B are and how far both initial temperatures are from unity. From Figures 3, 4 we can conclude that, in general, the inequality (31) is best satisfied if θ0Aθ0B.

3.2 Overshoot Mpemba effect

The emergence of the temperature overshoot described in Section 2.4 makes the crossover criterion employed in the SME become meaningless. Imagine that such a crossover takes place with TA(tc)=TB(tc)>1, but then TB(t) relaxes from above while TA(t) overshoots the steady-state value, TA(tO)=1. It is then possible that TA(t) relaxes (from below) later than TB(t). In that case, the initially hotter sample (A) would reach the steady state later than the initially colder sample (B), thus contradicting the existence of a ME, despite the crossover.

Reciprocally, imagine that TA(t) and TB(t) never cross each other but TB(t) overshoots the steady-state value and then relaxes from below. It is now possible that the initially hotter sample (A) reaches the steady state earlier than the initially colder sample (B), thus qualifying as a ME, despite the absence of any crossover. We will refer to this phenomenon as overshoot ME (OME) [44]. From the discussion in Section 2.4 we conclude that the OME requires εref>εcr(T0B,θ0B).

To characterize the existence of OME without a thermal crossover, we adopt the quantity [44].

DTtTt1lnTt.(32)

This quantity is (except for a factor) the Kullback–Leibler divergence of the Maxwellian VDF given by Eq. 24 (with Trot/Ttrθst) with respect to the steady-state Maxwellian. Note that D(T) is a positive-definite convex function of T. Therefore, we can define the OME by the crossover of D(TA) and D(TB) with, however, TA<1 and TA<TB.

In order to look for OME, the necessary condition for SME [given by Eq. 31] must be reversed. That is,

ΦT0A,θ0A>ΦT0B,θ0B.(33)

Establishing the most favorable conditions for OME is not as simple as just declaring the opposite of the SME condition. Firstly, we want for the colder sample to overshoot as much as possible the steady state, so that the relaxation from below is retarded maximally. This reasoning is translated into the condition of highly negative initial slope Φ(T0B,θ0B), which implies small θ0B (Figures 3, 4). On the other hand, two competing phenomena exist for the hotter sample: we want to either avoid any overshoot or force it to be as weak as possible, but we also want the relaxation to be faster than in the colder sample. Therefore, one needs θ0A > θ0B but, for very large values of θ0A, one might not find OME due to a slower relaxation of the hotter sample.

3.3 Initial preparation protocols

In order to study the absence or existence of ME, one needs to specify the initial conditions (T0A,θ0A) and (T0B,θ0B) of both samples. In previous studies [42, 44, 47] the values of θ0A and θ0B were freely chosen, without a specific reference to a previous protocol to initially prepare the samples.

Most of the interest of the present work resides in the proposal of protocols to generate the initial states of the samples involved in a ME experiment. The protocols are based on the proposed ST, and the initial states will be generated by assuming prior thermostat values (Tiwn,εi), i = A, B, and allowing both samples to reach their respective steady states (Tist,θist) before switching to the common posterior thermostat (Trefwn,εref) at t = 0. The values of the prior thermostats will be chosen to optimize the necessary conditions (31) and (33) for SME and OME, respectively.

According to Eqs. 26, 27, the ratios between the prior and posterior noise temperatures for desired values of T0i=Tist/Trefst, i = A, B, where Tist=TiwnT̃st(εi) and Trefst=TrefwnT̃st(εref), are

TiwnTrefwn=T0i2+θrefst2+θistγistγrefst2/3,i=A,B.(34)

3.3.1 Protocol for the standard Mpemba effect

In this case, we want to have θ0A << θ0B. According to Eqs. 26, and as observed in Figure 5, θst is an increasing function of ɛ and independent of Twn. Therefore, the most disparate values of θ0A and θ0B are obtained if the prior thermostats of samples A and B have ɛA = 0 and ɛB = 1, respectively. According to Figure 2, SME would be stronger and/or easier to find for lower values of β at fixed α and for lower values of α at fixed β. In addition, in order to define a cooling process, we need to choose proper values of TAwn/Trefwn and TBwn/Trefwn [Eq. 34], such that T0A=TAst/Trefst>T0B=TBst/Trefst>1. Finally, one must fix εref<min{εcr(T0A,θ0A),εcr(T0B,θ0B)} to prevent any possible overshoot. This minimum of the critical rotational-to-total noise intensity parameter is expected to be εcr(T0A,θ0A) because it corresponds to a more negative initial slope.

Thus, the designed protocol for SME reads as follows (Figure 8 for an illustrative scheme):

1) Start by fixing ɛA = 0 and ɛB = 1, in order to ensure θ0A < θ0B.

2) Choose TAwn/Trefwn and TBwn/Trefwn, such that T0A>T0B>Trefst.

3) Let both samples evolve and reach the steady states corresponding to their respective prior thermostats. These steady states will play the role of the initial conditions for our ME experiment.

4) Switch the values of the thermostats of both samples to a common reference pair of values (Trefwn,εref), such that no overshoot is expected, that is, εref<min{εcr(T0A,θ0A),εcr(T0B,θ0B)}. This thermostat switch fixes the origin of time, t = 0.

5) Finally, let both samples evolve and reach a common steady state.

FIGURE 8
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FIGURE 8. Scheme of the protocol for SME. (A) Choice of the splitting parameters ɛ. (B) Choice of the noise temperatures Twn.

For given (α, β), the numerical solutions of Eqs. 30 for different values of T0A and T0B—and with θ0A and θ0B given by Eqs. 28a, 28b, respectively—can be analyzed to determine whether SME is present or not. This provides the phase diagram presented in Figure 10A for ɛref = 0.1 and some pairs of coefficients of restitution.

3.3.2 Protocol for the overshoot Mpemba effect

In the OME case, it is convenient to have θ0Aθ0B, so that the adopted choices of ɛ for the prior thermostats are the reverse of those of SME, i.e., ɛA = 1 and ɛB = 0. Whereas in Section 3.2 we commented that the best situation for the OME is not always the opposite to that of the SME, the above choice helps us avoid or weaken a possible overshoot for the hotter sample. Again, in order to define a cooling process, we need to choose proper values of TAwn/Trefwn and TBwn/Trefwn [Eq. 34], and such that T0A>T0B>Trefst. Finally, εref>εcr(T0B,θ0B) to ensure overshoot of TB(t).

In analogy with the SME case, a protocol for observing OME is designed as follows (Figure 9):

1) Start by fixing ɛA = 1 and ɛB = 0, in order to ensure θ0A > θ0B.

2) Choose TAwn/Trefwn and TBwn/Trefwn, such that T0A>T0B>Trefst.

3) Let both samples evolve and reach the steady states corresponding to their respective prior thermostats. These steady states will play the role of the initial conditions for our ME experiment.

4) Switch the values of the thermostats of both samples to a common reference pair of values (Trefwn,εref), such that overshoot is ensured, that is, εref>εcr(T0B,θ0B). This thermostat switch fixes the origin of time, t = 0.

5) Finally, let both samples evolve and reach a common steady state.

FIGURE 9
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FIGURE 9. Same as in Figure 8, but for OME.

Figure 10B shows a phase diagram for the occurrence of OME (with ɛref = 0.9) for the same pairs of coefficients of restitution as before.

FIGURE 10
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FIGURE 10. Phase diagrams in the plane T0A*/T0B* vs. T0B* for the emergence of (A) SME (with ɛref = 0.1) and (B) OME (with ɛref = 0.9). Four different pairs of the coefficients of restitution are considered: (α, β) = (0.7, − 0.7), (0.7, 0), (0.9, − 0.7), and (0.9, 0). The insets show magnified views of the indicated regions.

3.4 Comparison with simulation results

In order to check the initialization protocols for detecting SME and OME, we have run DSMC and EDMD simulations. The simulation details are the same as introduced in Section 2.3. Simulation points correspond to the average over ensembles of 100 replicas. Again, no instabilities were observed.

3.4.1 Standard Mpemba effect

Figure 11 presents results for the SME protocol introduced in Section 3.3.1. As we can observe, the theoretical predictions agree very well with DSMC and EDMD simulation data.

FIGURE 11
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FIGURE 11. Time evolution of the reduced mean temperature T* and the rotational-to-translational temperature ratio θ for SME initialization protocol, as described in Section 3.3.1. (A,B): (α, β) = (0.7, 0) and ɛref = 0.1; (C,D): (α, β) = (0.7, − 0.7) and ɛref = 0.1; (E,F); (α, β) = (0.9, 0) and ɛref = 0.6; and (G,H): (α, β) = (0.9, − 0.7) and ɛref = 0.1. Thick and dashed lines correspond to the theoretical prediction from Eq. 30, dotted lines represented the steady state value, and symbols refer to DSMC and EDMD simulation results. The inset in panel (E) shows the evolution of the temperature difference TA*TB*.

The chosen initial temperature conditions are (T0A,T0B)=(3,2) for Figure 11A, (T0A,T0B)=(4,2) for Figures 11C,G, and (T0A,T0B)=(3,2.92) for Figure 11E. Moreover, ɛref = 0.1 for all cases, except for the case of Figures 11E,F, that is chosen to be ɛref = 0.6. The values of TAwn/Trefwn and TBwn/Trefwn are given in each case by Eq. 34 to ensure the desired initial temperatures. All these cases avoid overshoot, in agreement with the case θ0 = θst(ɛ = 0) shown in Figure 7. Moreover, the cases with ɛref = 0.1 are inside the region where SME is predicted to be present in Figure 10A.

The value ɛref = 0.6 for the system (α = 0.9, β = 0) was chosen instead of ɛref = 0.1 to avoid the need of taking very high initial temperatures (Figure 10A) and also to prevent overshoot (Figure 7). The price paid for this choice of coefficients of restitution is that the difference θ0Aθ0B is not too high, see Figures 2, 11F. Then, initial temperature values are restricted to be very similar and the crossing characterizing the SME is much less pronounced than in the other cases. However, the inset in Figure 11E shows a well defined change of sign of the difference TATB.

3.4.2 Overshoot Mpemba effect

The theoretical results stemming from the OME protocol introduced in Section 3.3.2 are compared with simulations in Figure 12, again with an excellent agreement.

FIGURE 12
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FIGURE 12. Time evolution of the reduced mean temperature T*, the entropy-like quantity D, and the rotational-to-translational temperature ratio θ for OME initialization protocol described in Section. 3.3.2 with ɛref = 0.9. (AC): (α, β) = (0.7, 0); (DF): (α, β) = (0.7, − 0.7); (GI); (α, β) = (0.9, 0); and (JL): (α, β) = (0.9, − 0.7). Thick and dashed lines correspond to the theoretical prediction from Eqs. 30, dotted lines represented the steady state value, and symbols refer to DSMC and EDMD simulation results.

The initial temperatures are (T0A,T0B)=(1.22,1.1) for Figure 12A, (T0A,T0B)=(2,1.5) for Figures 12D,J, and (T0A,T0B)=(1.2,1.1) for Figure 12G. Moreover, a common value ɛref = 0.9 is chosen to ensure that overshoot appears for the colder samples, as depicted in Figure 7, and that OME is expected from the phase diagram shown in Figure 10B.

Whereas no crossing between TA and TB is observed in Figure 12, such a crossing is present between the entropy-like quantities DA and DB [Eq. 32]. As explained in Section 3.2, this characterizes the OME, where the initially colder system relaxes later to the steady state.

4 Concluding remarks

In this work, we have studied the homogeneous states of a dilute granular gas made of inelastic and rough hard disks lying on a two-dimensional plane. The inelasticity and roughness are mathematically described by constant coefficients of normal (α) and tangential (β) restitution, respectively. In order to avoid frozen long-time limiting states, the disks are assumed to be heated by the stochastic force- and torque-based ST. This novel stochastic thermostat injects energy to both translational and rotational degrees of freedom. Each specific thermostat of this type is univocally determined by its associated noise temperature, Twn ≥ 0 [Eq. 10], and the rotational-to-total noise intensity, 0 ≤ ɛ ≤ 1 [Eq. 9].

The system is assumed to be fully described by the instantaneous one-particle VDF, its dynamics being then given by the BFPE, Eq. (11). It is known that the steady-state VDF is not a Maxwellian, as occurs for ɛ = 0 [75], and this is even more the case with the instantaneous transient VDF. However, these nonGaussianities are expected to be small enough as to approximate the VDF by a Maxwellian, as previously done for ɛ = 0 [47]. Therefore, we have worked under the two-temperature MA introduced in Section 2.2, which allows us to account for the dynamics of the systems just in terms of the reduced total granular temperature, T, and the rotational-to-translational temperature ratio, θ, according to Eqs. 30. The steady-state values can be explicitly expressed in terms of the mechanical properties of the disks, Eqs. 26. Steady and transient states predicted by the MA are tested via DSMC ans EDMD, with very good agreement, as observed in Figures 5, 6. This reinforces the validity of our approach. As expected, θst is an increasing function of ɛ and independent of Twn, whereas T̃stTst/Twn can be a nonmonotonic function of ɛ (Figure 5).

The main core of this work has been the description of ME in cooling processes in this system, with special emphasis on the elaboration of preparation protocols for the generation of the initial states. We noted that, if T − 1 does not change its sign during the evolution, the usual form of ME, SME, can emerge if θ0Aθ0B, A being the initially hotter sample. This SME is characterized by a single crossing (or, in general, an odd number of them) between the temperature curves, thus inducing that the initially colder system, B, relaxes more slowly toward the final steady state. However, we have realized that an overshoot (or change of sign of T − 1) might appear if the rotational-to-total noise intensity is larger than a certain critical value εcr(T0,θ0), as shown for some cases in Figure 7. Therefore, the SME is no longer ensured and, following a recent work [44], ME might appear without a crossing between TA and TB but with a crossing between the entropy-like quantities DA and DB [see Eq. 32]. This peculiar type of ME is termed OME and is described in Section 3.2. According to the MA, the most favorable condition for the OME changes from the SME one to θ0Aθ0B.

Protocols for generating initial conditions to observe both SME and OME have been presented in Sections 3.3.1, 3.3.2, respectively. While reminiscent of the protocols previously considered in the case of sheared inertial suspensions [53], the protocols proposed here represent novel instructions to elaborate a ME experiment in homogeneous states of granular gaseous systems. We have based those protocols on the steady states of the ST, taking advantage of the increase of θst as ɛ increases. Thus, to guarantee the biggest possible difference |θ0Bθ0A| we have fixed prior thermalization processes with (ɛA, ɛB) = (0, 1) for SME and (ɛA, ɛB) = (1, 0) for OME. Moreover, in this prior thermalization stage, TAwn/Trefwn and TBwn/Trefwn, where (Trefwn,εref) characterizes the posterior thermostat, are chosen to fine-tune both T0A and T0B. It is crucial to choose εref<min{εcr(T0A,θ0A),εcr(T0B,θ0B)} for SME to avoid overshoot, and εref>εcr(T0B,θ0B) for OME to ensure the overshoot of the initially colder sample. These protocols, together with the MA predictions, allowed us to elaborate phase diagrams for SME and OME emergences, as depicted in Figure 10.

The theoretical descriptions of the SME and OME initialization protocols have been tested by DMSC and EDMD simulations, finding a very good agreement between theory and simulations, as observed in Figures 11, 12, respectively.

One can then conclude that, despite its simplicity, the MA captures very well the dynamics for this system. In turn, this implies that the SME and OME protocols for the initial-state preparation described in this paper are trustworthy.

We expect that this work will be useful to the ME community in the search for practical protocols able to generate adequate initial states. Whereas this work is restricted to a specific physical model, the phenomenology that arises might be general in Mpemba-like effects in out-of-equilibrium statistical mechanical systems. In fact, the overshoot mechanism introduced has been previously observed in other gaseous models [44], and it is somehow reminiscent of the supercooling hypothesis for the ME in water [19]. Furthermore, the two-parameter-based protocols proposed in this paper might inspire the development of protocols for Mpemba-like effects in other systems.

To sum up, the designs of the protocols can be based on a very obvious statement: generate the initial conditions from steady states for specific values of the parameters of the thermostat; then, change the values of those thermostat parameters and let the system relax to the new steady state. In general, if the temperature (or the energy) of the physical system in a certain model is coupled to n independent control variables (i.e., the dynamics is described by n + 1 differential equations), one should look for a proper choice of n + 1 parameters of the model that will determine directly the value of each variable at the steady state.

Finally, given the simplicity of the hard-disk system studied in this paper, we hope it can be experimentally realizable, thus opening up the possibility of reproducing the ME by an adequate control of the external forcing mechanisms.

Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: https://github.com/amegiasf/MpembaSplitting.

Author contributions

AM and AS contributed to the conception and design of the study. AM performed the computer simulations and wrote the first draft of the manuscript. Both authors contributed to manuscript revision, read, and approved the submitted version.

Funding

The authors acknowledge financial support from Grant No. PID2020-112936GB-I00 funded by MCIN/AEI/10.13039/501100011033, and from Grants No. IB20079 and No. GR21014 funded by Junta de Extremadura (Spain) and by ERDF “A way of making Europe.” AM is grateful to the Spanish Ministerio de Ciencia, Innovación y Universidades for a predoctoral fellowship FPU2018-3503.

Acknowledgments

The authors are grateful to the computing facilities of the Instituto de Computación Científica Avanzada of the University of Extremadura (ICCAEx), where the simulations were run.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: granular gases, kinetic theory, Mpemba effect, direct simulation Monte Carlo, molecular dynamics

Citation: Megías A and Santos A (2022) Mpemba-like effect protocol for granular gases of inelastic and rough hard disks. Front. Phys. 10:971671. doi: 10.3389/fphy.2022.971671

Received: 17 June 2022; Accepted: 19 July 2022;
Published: 06 October 2022.

Edited by:

Ramon Planet, University of Barcelona, Spain

Reviewed by:

Stefano Mossa, Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA), France
Sutapa Roy, Indian Institute of Technology Gandhinagar, India

Copyright © 2022 Megías and Santos. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Alberto Megías, albertom@unex.es

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