PHYSICAL REVIEW E 99, 060901(R) (2019)
Rapid Communica ions
La ge Mpemba-like e ec in a gas o inelas ic ough ha d sphe es
Au o a To en e,1Miguel A. López-Cas año,2An onio Lasan a,1F ancisco Vega Reyes,2An onio P ados,3,*
and And és San os2
1G ego io Millán Ins i u e o Fluid Dynamics, Nanoscience and Indus ial Ma hema ics, Depa men o Ma e ials Science and Enginee ing
and Chemical Enginee ing, Uni e sidad Ca los III de Mad id, 28911 Leganés, Spain
2Depa amen o de Física and Ins i u o de Compu ación Cien í ica A anzada (ICCAEx), Uni e sidad de Ex emadu a, 06006 Badajoz, Spain
3Física Teó ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain
(Recei ed 26 Feb ua y 2019; published 6 June 2019)
We epo he eme gence o a gian Mpemba e ec in he uni o mly hea ed gas o inelas ic ough ha d sphe es:
The ini ially ho e sample may cool soone han he colde one, e en when he ini ial empe a u es di e by mo e
han one o de o magni ude. In o de o unde s and his beha io , i su ices o conside he simples Maxwellian
app oxima ion o he eloci y dis ibu ion in a kine ic app oach. The la geness o he e ec s ems om he ac
ha he o a ional and ansla ional empe a u es, which obey wo coupled e olu ion equa ions, a e compa able.
Ou heo e ical p edic ions ag ee e y well wi h molecula dynamics and di ec simula ion Mon e Ca lo da a.
DOI: 10.1103/PhysRe E.99.060901
Le us conside wo beake s o wa e a di e en empe a-
u es. Mpemba and Osbo ne showed ha he ini ially ho e
sample cools soone unde ce ain condi ions [1], i.e., he
cu e gi ing he ime e olu ion o i s empe a u e c osses ha
o he ini ially coole sample and s ays below i o longe
imes. This is called he Mpemba memo y e ec , which is
known since an iqui y in cul u es o which wa e in he o m
o ice and snow is common [2]. La e , he Mpemba e ec
has been clea ly iden i ied in di e en physical sys ems [3–7],
al hough he e is s ill some deba e abou i s exis ence in wa e
[8].
F om a physical poin o iew, one would like o answe
how di e en he ini ial p epa a ion o wo samples o he
sys em unde s udy mus be so ha he Mpemba e ec a ises.
This is he main—cu en ly un esol ed in gene al—ques ion,
al hough he e has been some ecen p og ess in his espec
[5,6]. Lu and Raz [5] analyzed he Mpemba e ec in a gene ic
Ma ko ian sys em by moni o ing he elaxa ion o an en opy-
like a iable ha measu es he dis ance o he s eady s a e.
This makes i possible o de ine and in es iga e Mpemba-like
e ec s in sys ems o which he e is no an ob ious de ini ion
o a nonequilib ium empe a u e, bu makes he compa ison
wi h he usual expe imen al se up desc ibed abo e di icul .
A di e en app oach was ca ied ou by some o us in he
s udy o he Mpemba e ec o a g anula luid o smoo h ha d
sphe es [6]. The ein, he g anula empe a u e—basically he
a e age kine ic ene gy pe pa icle—is he physical quan i y
moni o ed o in es iga e he Mpemba e ec . In he smoo h-
sphe e case, he angula eloci ies play no ole since he e is
no ene gy ans e be ween he ansla ional and he o a ional
deg ees o eedom, and he kine ic ene gy is hus pu ely
ansla ional. We showed ha he Mpemba e ec s ems om
he coupling o he g anula empe a u e and he ku osis,
which measu es he de ia ion o he eloci y dis ibu ion
unc ion om he Maxwellian shape a he lowes o de . Mo e
*Co esponding au ho : [email p o ec ed]
speci ically, i is he di e ence be ween he ini ial alues o
he ku osis o he wo samples ha con ols he appea ance
o he Mpemba e ec .
In he g anula luid o smoo h ha d sphe es, he ku osis is
ypically small. On he one hand, his acili a es he heo e ical
analysis, because i makes i possible o linea ize he e olu ion
equa ions and hus gi e a quan i a i e p edic ion o how
di e en he ini ial ku oses mus be o enable he Mpemba
e ec . On he o he hand, he smallness o he ku osis limi s
he magni ude o he Mpemba e ec : The ini ial empe a u es
mus be qui e close; see Eq. (5) and Fig. 1(b) o Re . [6].
I has e y ecen ly been shown ha a di e en memo y
e ec , he Ko acs e ec [9–13], is much la ge and mo e
complex in a g anula gas o ough sphe es [14] han in
he smoo h-sphe e case [15,16]. Wha makes i possible o
unde s and he la geness o he Ko acs e ec is he coupling
be ween he ansla ional and o a ional empe a u es, which
a e o he same o de o magni ude. In addi ion, he basic
ea u es o he memo y e ec can be unde s ood wi hin he
Maxwellian (Gaussian) app oxima ion, wi hou ha ing o e-
so o highe o de cumulan s.
The abo e pic u e p omp s us o look in o he Mpemba e -
ec in a luid o ough inelas ic ha d sphe es. Rema kably, we
show in he ollowing ha he Mpemba e ec can be explained
wi hin a Gaussian amewo k, bo h quali a i ely and quan i-
a i ely. Physically speaking, he coupling be ween he o a-
ional and ansla ional deg ees o eedom and hus he ex-
is ence o wo compa able bu di e en empe a u es su ices
o explain he memo y e ec . Mo eo e , we gi e a pic u e o
he unde lying physical condi ions and discuss he possible
ele ance o he wo- empe a u e mechanism in o he sys ems.
The e o e, le us conside a dilu e gas o inelas ic ough
ha d sphe es, wi h mass m, diame e σ, and momen o
ine ia I. Hence o h, we employ he dimensionless momen
o ine ia κ≡4I/mσ2, and i s speci ic alue o uni o m
solid sphe es (κ=2
5) whene e a de ini e alue is needed.
T ansla ional and angula pa icle eloci ies a e deno ed as
and ω, espec i ely.
2470-0045/2019/99(6)/060901(6) 060901-1 ©2019 Ame ican Physical Socie y
AURORA TORRENTE e al. PHYSICAL REVIEW E 99, 060901(R) (2019)
Collisions be ween mac oscopic pa icles a e inelas ic, i.e.,
ene gy is no conse ed [17]. Fo he inelas ic ough ha d
sphe e model, a bina y collision is cha ac e ized by wo pa-
ame e s: he coe icien o no mal es i u ion 0 ⩽α⩽1 and
he coe icien o angen ial es i u ion −1⩽β⩽1[18–20].
They a e in insic p ope ies o he ma e ial and de e mine
he sh inking o he no mal and angen ial componen s o he
ela i e eloci y o he wo su ace poin s a con ac [21]. The
collisional model based on hese wo pa ame e s is su icien ly
accu a e in a a ie y o ma e ials [17], wi h mos o hem
p esen ing expe imen al alues in he in e als α∈(0.7,0.95)
and β∈(−0.5,0.5) [22]. Consis en ly, he analysis ca ied
ou in his pape ocuses on his egion o he (α,β) plane.
Addi ionally, ene gy is homogeneously injec ed o he
ansla ional deg ees o eedom o all he g ains by a s ochas-
ic he mos a Fmodeled as a Gaussian whi e noise, i.e.,
Fi( )=0,Fi( )Fj( )=Im2χ2
0δijδ( − ), whe e i,j e-
e o he pa icles’ indexes, Iis he 3 ×3 uni ma ix, and χ2
0
gi es he “s eng h” o he s ochas ic o cing [23–28].
He e, we p o ide he minimal heo e ical amewo k
needed o he unde s anding o he Mpemba e ec in he
g anula gas (see Re s. [29–31] o a de ailed accoun o he
kine ic heo y calcula ions). The dynamics o ou sys em is
go e ned by he inelas ic Bol zmann-Fokke -Planck equa ion
o he single-pa icle eloci y dis ibu ion unc ion ( ,ω, )
[27]. F om he kine ic equa ion, he e olu ion equa ions o
he a e age quan i ies o in e es a e de i ed [29,30].
We es ic ou sel es o homogeneous and iso opic s a es,
o which  =0and ω=0. The basic physical in o ma-
ion is hus encoded in he ansla ional and o a ional g anula
empe a u es T =m
3 2and T =I
3ω2. Al e na i ely, he
same in o ma ion is p o ided by he empe a u e a io θand
he o al empe a u e T,
θ( )=T ( )
T ( ),T( )=T ( )+T ( )
2.(1)
The g anula gas is inhe en ly a nonequilib ium sys-
em and, he e o e, equipa i ion is b oken, i.e., θ= 1.
Thus, he simples desc ip ion o he ough-sphe e g an-
ula gas is p o ided by he Maxwellian app oxima ion,
in which he ollowing bi a ia e Gaussian o m is as-
sumed o he eloci y dis ibu ion unc ion, ( ,ω, )≃
n[mI/4π2T ( )T ( )]
3
2exp [−m 2
2T ( )−Iω2
2T ( )], whe e nis he
numbe densi y.
In he long- ime limi , he g anula gas eaches a s eady
s a e due o he ac ion o he s ochas ic o ce. This s eady s a e
is comple ely cha ac e ized by θs and Ts in he Maxwellian
app oxima ion. Thei exp essions in e ms o he coe icien s
o es i u ion and he s ochas ic o cing in ensi y a e [30]
θs =1+β
2+κ−1(1 −β),Ts =1+θs
23m3/2χ2
0
4√πnσ2γs 2/3
,
(2a)
γs ≡1−α2+2(1 −β2)
2+κ−1(1 −β).(2b)
No e ha θs ⩽1 is independen o αin he Maxwellian
app oxima ion bu highe o de app oxima ions in oduce a—
a he weak—dependence on α[30].
I is use ul o in oduce dimensionless a iables o em-
pe a u e and ime. Then, we de ine T∗≡T/Ts and ∗≡
2nσ2πTs
/m . In he Maxwellian app oxima ion, T∗and θ
e ol e acco ding o [31]
∂ ∗ln T∗=(T∗,θ),∂
∗ln θ=(T∗,θ),(3)
wi h he de ini ions
(T∗,θ)=1(T∗)+2(T∗,θ)+3(T∗,θ),(4a)
(T∗,θ)=−(1 +θ)(T∗,θ)−3(T∗,θ)
θ(1 −θs ).(4b)
Abo e, we ha e in oduced he no a ion
1≡2
3
γs
T∗(1 +θs ),
2≡−2
3T∗(1 +θs )
(1+θ)3γs ,(5a)
3≡2
3KT∗(1 +θs )
(1+θ)31−θ
θs (1 −θs ),(5b)
wi h K≡κ(1 +β)2/(1 +κ)2[32]. No e ha he ime e olu-
ion o he empe a u e is go e ned by he unc ion , which
does no only depend on T∗; his is a necessa y condi ion o
he appea ance o he Mpemba e ec .
Imagine wo ini ial s a es (T∗
0A,θ
0A) and (T∗
0B,θ
0B), wi h
T∗
0A>T∗
0B>1, o he same g anula gas, i.e., wi h he same
alues o he coe icien s o es i u ion αand β. Le us deno e
by T∗
A( ∗) and T∗
B( ∗) he associa ed decays o he empe a u e
o he s eady s a e: A Mpemba-like e ec is b ough abou
when he e exis s a c ossing ime ∗
×such ha T∗
A( ∗)<T∗
B( ∗)
o ∗> ∗
×.
A necessa y—and physically in ui i e—condi ion o ha -
ing he Mpemba e ec is ha he ini ially ho e sample cools
as e han he coole one o sho imes, when he sys em
s ill keeps “memo y” o i s ini ial condi ions and is in he i s
s age o he so-called kine ic egime. Fo sho enough imes,
we can conside ha he sys em is exponen ially cooling wi h
a cha ac e is ic a e oughly equal o he ini ial alue o −
[33] and hen a necessa y condi ion o he Mpemba e ec o
be p esen is
(T∗
0A,θ
0A)<(T∗
0B,θ
0B).(6)
Le us in es iga e he beha io o (T∗,θ)asa unc ion
o θ, o ixed T∗, o unde s and unde which condi ions he
Mpemba e ec is expec ed. The e a e h ee dis inc e ms in
: (i) he i s one, 1(T∗), is a hea ing e m ha s ems om
he s ochas ic o cing and is hus independen o θ, (ii) he
second one, 2(T∗,θ), is he ypical cooling e m o g anula
gases, which is also p esen o smoo h sphe es [34], and
(iii) he hi d one, 3(T∗,θ), is a pu ely oughness e m (no e
ha K=0 o β=−1) and hea s (cools) he sys em when
θ<θ
s (θ>θ
s ). The sign and magni ude o (T∗,θ) esul s
om he compe i ion among hose h ee e ms.
In ligh o he abo e, we analyze he beha io o 2and
3as a unc ion o θ, o ixed T∗. While he cooling e m
2is a mono onically inc easing unc ion o θ[35], he
oughness e m 3shows a mo e complex beha io . S a ing
om θ=0+,3 i s dec eases wi h inc easing θ, anishes a
060901-2
LARGE MPEMBA-LIKE EFFECT IN A GAS OF … PHYSICAL REVIEW E 99, 060901(R) (2019)
FIG. 1. Densi y plo s o (T∗,θ) as de ined by Eq. (4a). Two
ep esen a i e examples o he coe icien o es i u ion, (a) α=1
and (b) α=0.9, a e conside ed, o β=0. The con ou lines (solid
o nega i e , dashed o posi i e ) a e sepa a ed by an amoun
 =0.05. The hick solid line is he locus (T∗,θ)=0 and he
ci cle ma ks he s eady-s a e poin (T∗,θ)=(1,θs ).
θ=θs , eaches a (nega i e) minimum alue a θ=2+3θs ,
and inally ends o ze o om below in he limi θ→∞.
The o e all beha io o 2+3as a unc ion o θdepends
on he alues o he coe icien s o es i u ion (α,β). Bo h
2and 3g ow wi h inc easing θbeyond he minimum o
3, i.e., o θ>2+3θs . On he o he hand, as αapp oaches
uni y, he decay o 3 o small θdomina es o e he g ow h
o 2, esul ing in a nonmono onic dependence o on θ.
This is illus a ed in Fig. 1(a), which pu s o wa d a densi y
plo o as a unc ion o (T∗,θ) o uni o m solid sphe es
in he limi ing case (α=1,β =0). A nonmono onic beha -
io is nea ly obse ed, especially o he igh o he locus
(T∗,θ)=0, i.e., whe e <0 and he sys em cools. As α
is dec eased, he magni ude o he cooling e m 2inc eases,
e en ually becoming he dominan one o small θi 1 −αis
la ge enough. The ein, a mono onically inc easing beha io is
obse ed, as illus a ed in Fig. 1(b) o (α=0.9,β =0).
To ca y ou a mo e quan i a i e analysis, we impose ha
∂(T∗,θ)/∂θ|
θ=0, which leads o

θ(α,β)=2−3κ−3(1 +κ)1−α2
1−β2,(7)
and s udy he sign o 
θ[36]. I 
θ⩽0, he e is no physically
meaning ul minimum and is a mono onically inc easing
unc ion o θ.I 
θ>0, displays a minimum a θ=
θ. Equa-
ion (7)implies ha 
θ>0i αis su icien ly close o uni y and
|β|is su icien ly small. This is consis en wi h he quali a i e
discussion abo e, and i is illus a ed in Fig. 2. The ein, he
locus
θ=0 sepa a es he egions inside which is mono onic
(below i ) and nonmono onic (abo e i ). On he one hand,

θ⩽0 o all βwhen α⩽αc=√(1 +6κ)/3(1 +κ), which
gi es αc=√17/21 ≃0.9 o uni o m solid sphe es. On he
o he hand,
θ>0 o α>α
conly i β2<1−3(1 −α2)(1 +
κ)/(2 −3κ)[37].
Fo he sake o conciseness, we es ic ou sel es o he
simple mono onic si ua ion α⩽αcin he emainde o he
pape . Le us look again a Fig. 1(b), in which he limi ing—
less a o able—case (α=αc,β =0) is shown. As al eady
s a ed be o e, we conside wo poin s, A≡(T∗
0A,θ
0A) and B≡
(T∗
0B,θ
0B) wi h T∗
0A>T∗
0B, co esponding o di e en ini ial
FIG. 2. Locus 
θ(α,β)=0in he(β,α) plane. We ha e a non-
mono onic beha io o (T∗,θ) sθwi h a minimum a 
θ>0
abo e he cu e, whe e 
θisgi enbyEq.(7), whe eas (T∗,θ)has
a mono onically inc easing beha io below he cu e. He e, κ=2
5
(uni o m solid sphe es).
condi ions, o s udy he Mpemba e ec . Since is mono onic,
i su ices o ake θ0A<θ
0B(i.e., he ini ially ho e sys em
has i s kine ic ene gy mo e concen a ed in he ansla ional
modes han he ini ially coole one) o ul ill Eq. (6). Equa ion
(6), howe e , is no a su icien condi ion o he Mpemba
e ec o appea : The dispa i y be ween (T∗
0A,θ
0A) and
(T∗
0B,θ
0B) mus be la ge enough, because he elaxa ion is
no pu ely exponen ial. This en ails ha he dispa i y be ween
θ0Aand θ0Bmus be also la ge enough. This is illus a ed in
Fig. 3(a), whe e we analyze he eme gence o he Mpemba
e ec o se e al pai s o he ini ial empe a u es (T∗
0A,T∗
0B).
Fo each one o hese pai s, he lines in he (θ0B,θ
0A) plane
delimi ing he egions wi h and wi hou he Mpemba e ec
a e plo ed.
Fo he eme gence o he Mpemba e ec , he mos a o -
able si ua ion is ha ing he kine ic ene gy comple ely concen-
a ed in (i) he ansla ional deg ees o eedom o he highe
FIG. 3. Phase diag ams o he Mpemba e ec o α=0.9and
β=0. In panel (a), we conside he (θ0B,θ
0A) plane: The Mpemba
e ec is p esen (absen ) o poin s below (abo e) he plo ed
lines, which co espond o se e al choices o he ini ial condi ions;
speci ically, (i) (T∗
0A,T∗
0B)=(5,4), (ii) (T∗
0A,T∗
0B)=(4,3), and (iii)
(T∗
0A,T∗
0B)=(5,3). No e he di e en scales in bo h axes. In panel
(b), we plo he line in he (T∗
0B,T∗
0A/T∗
0B) plane below which he
Mpemba e ec may appea , p o ided ha he kine ic ene gy o he
ini ially ho e (colde ) sample is concen a ed in he ansla ional
( o a ional) deg ees o eedom o a su icien ex en .
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AURORA TORRENTE e al. PHYSICAL REVIEW E 99, 060901(R) (2019)
1
2
3
4
5α = 0.9, β = −0.5
MpAB = 0.54
MpAC = 0.39
(a)
1
2
3
4
5α = 0.9, β = 0
MpAB = 0.29
MpAC = 0.16
(c)
024681012
1
2
3
4
5α = 0.9, β = 0.5
MpAB = 0.16
MpAC = 0.02
(e)
α = 0.7, β = −0.5
MpAB = 1.20
MpAC = 0.93
(b)
α = 0.7, β = 0
MpAB = 0.82
MpAC = 0.59
(d)
024681012
α = 0.7, β = 0.5
MpAB = 0.65
MpAC = 0.38
( )
*
T / T
s
FIG. 4. Mpemba e ec in he g anula gas o ough ha d sphe es. The elaxa ion o he scaled empe a u e T∗ o i s s eady-s a e alue is
shown o di e en alues o he coe icien s o es i u ion αand β. The MD da a (open ci cles) and, especially, he DSMC da a ( iangles) a e
in e y good ag eemen wi h he heo e ical alues (lines). Dashed blue, solid g een (ligh g ay), and solid ed lines e e o ini ial s a es gi en
by (T∗
0A,θ
0A)=(5,0.01), (T∗
0B,θ
0B)=(4,10), and (T∗
0C,θ
0C)=(3,100), espec i ely. The Mpemba e ec is nea ly obse ed, wi h he ini ially
ho es sys em being he i s one o each he s eady s a e.
empe a u e T∗
0A, i.e., θ0A=0, and (ii) he o a ional deg ees
o eedom o he lowe empe a u e T∗
0B, i.e., θ0B→∞.
This limi ing case is conside ed in Fig. 3(b). The ange o
ini ial empe a u e a ios T∗
0A/T∗
0Bleading o he eme gence o
he Mpemba e ec inc eases (almos exponen ially) wi h he
colde ini ial empe a u e T∗
0B. Fo ins ance, i T∗
0B=15.3, T∗
0A
can be 100 imes la ge han T∗
0Band s ill he Mpemba e ec is
obse ed. In all he cases plo ed in Fig. 3, we ha e conside ed
ha he Mpemba e ec is p esen when he c ossing ime
∗
×—i i e e exis s—is smalle han 10 [38].
To check he accu acy o ou heo e ical p edic ions, we
ha e ca ied ou molecula dynamics (MD) and di ec Mon e
Ca lo (DSMC) simula ions [31]. Di e en sys ems a e con-
side ed by a ying he alues o αand β. Fo each pa icula
pai (α, β), he sys em is ini ialized a h ee poin s A,B,
and Cwi h (T∗
0A,θ
0A)=(5,0.01), (T∗
0B,θ
0B)=(4,10), and
(T∗
0C,θ
0C)=(3,100), espec i ely. Acco ding o ou heo e -
ical amewo k, he Mpemba e ec is expec ed o eme ge in
he cases (A,B) and (A,C), bu no in he case (B,C).
Figu e 4shows he ime e olu ion o he g anula empe -
a u e. The cu es co espond o α=0.7 and 0.9, and β=
−0.5, 0, and 0.5, bu he quali a i e pic u e explained below
emains unal e ed o o he choices, as long as α⩽αc≃
0.9. The Mpemba e ec is clea ly obse ed as he ini ially
ho es sample (A) is he as es one o each he s eady s a e.
Mo eo e , in all he cases, he h ee plo ed cu es— he MD
da a (ci cles), he DSMC da a ( iangles), and he Maxwellian
heo y (line)—p esen a ema kably excellen ag eemen .
Quali a i ely, he s eng h o he Mpemba e ec — he
sepa a ion be ween he wo elaxa ion cu es o imes
longe han ∗
×—inc eases as αdec eases a ixed β(highe
inelas ici y) and as βgoes o smalle alues a ixed α(lowe
060901-4
LARGE MPEMBA-LIKE EFFECT IN A GAS OF … PHYSICAL REVIEW E 99, 060901(R) (2019)
oughness). Thus, quan i a i ely, we can measu e he magni-
ude o he Mpemba e ec by de ining a Mpemba pa ame e
MpAB as he ex emum o he di e ence o dimensionless
empe a u es T∗
B( )−T∗
A( ) o ∗> ∗
×. The alues o MpAB
and MpAC ob ained om he solu ion o Eq. (3)a eshownin
each one o he panels o Fig. 4and a e consis en wi h he
quali a i e beha io desc ibed abo e. Fo ixed oughness β,
he mo e inelas ic he sys em is, he la ge Mp becomes. Fo
ixed inelas ici y α, he smoo he he sys em is, he la ge Mp
becomes. No e ha he alues o Mp a e ypically o he o de
o uni y o ou sys em o ough sphe es. In he smoo h-sphe e
case, howe e , Mp is much smalle , being ypically o he
o de o a ew housand hs o he cases epo ed in Re . [6].
While o smoo h sphe es he in luence o he ku osis
on he e olu ion o T∗is e y weak, a s ong impac o he
o a ional- ansla ional pa i ion is p esen in he case o ough
sphe es.
We ha e nea ly obse ed he Mpemba e ec in he gas o
inelas ic ough ha d sphe es. I s ems om he coupling in
he e olu ion o he o a ional and ansla ional empe a u es,
which a e, in gene al, o he same o de , θ=O(1). Fo low
enough α, i.e., la ge enough inelas ici y, we ha e shown
ha he mo e concen a ed in he ansla ional deg ees o
eedom he o al kine ic ene gy is, he as e he sys em cools.
The e o e, he ini ially ho e sys em mus ha e i s kine ic
ene gy mo e concen a ed in he ansla ional modes han he
ini ially coole one o acili a e he Mpemba e ec . Mo eo e ,
we ha e quan i ied how la ge his concen a ion mus be. In
he ough-sphe e case, he ini ial empe a u es o he wo
samples can be qui e di e en and de ini ely do no need o be
close o each o he , as happens in he smoo h-sphe e g anula
gas.
The s eng h o he e ec has been quan i ied by he
Mpemba pa ame e Mp de ined as he maximum sepa a ion—
ela i e o he s a iona y empe a u e—be ween he elaxa ion
cu es a e he c ossing. The Mpemba e ec in he ough-
sphe e g anula gas is eally huge, wi h ypical o de o uni y
alues o Mp. This has o be con as ed wi h he Mpemba
e ec in he smoo h-sphe e case ha , al hough dis inc ly
obse ed in Re . [6], is qui e small (Mp ∼10−3).
Also, i is wo h emphasizing ha he Mpemba e ec can
be explained wi h a minimal Gaussian app oxima ion o he
dis ibu ion unc ion. The la geness o he e ec shows ha
he sys em is sweeping a - om-equilib ium s a es bu , in e -
es ingly, hese s a es can be desc ibed by a so o “local equi-
lib ium” app oxima ion bu wi h wo dis inc empe a u es.
This is a di e ence wi h he hyd odynamic egime, in which
he o a ional empe a u e becomes ensla ed o he o al one
o long enough imes, i.e., θbecomes ime independen o e
he hyd odynamic scale and Tis he ele an hyd odynamic
ield.
In he ough-sphe e case, he co ec ions in oduced by he
cumulan s a e expec ed o emain small [39]. This expec a ion
is suppo ed by he excep ionally good ag eemen ound in
Fig. 4be ween he Maxwellian analy ical p edic ions and he
DSMC da a—which gi e he nume ical in eg a ion o he
kine ic equa ion. In addi ion, he ag eemen o ou heo y wi h
MD simula ions ells us ha ou kine ic desc ip ion holds o
he low-densi y gas.
The esul s in he p esen wo k sugges ha he wo-
empe a u e mechanism ound he e may be signi ican o ob-
se a ion o a huge—and hus expe imen ally measu able—
Mpemba e ec in a a ie y o sys ems wi h se e al dis inc
empe a u es, such as s uc u al glasses [40,41], me als unde -
going plas ic de o ma ion [42], and g anula mix u es [43].
This is enough o b ing o he o e he Mpemba e ec , which
may e en be enhanced i o he a iables like highe o de
cumulan s a e also ele an .
This wo k has been suppo ed by he Spanish Minis-
e io de Ciencia, Inno ación y Uni e sidades and Agencia
Es a al de In es igación G an s (pa ially inanced by he
ERDF) No. MTM2017-84446-C2-2-R (A.L. and A.T.), No.
FIS2016-76359-P (F.V.R. and A.S.), No. FIS2017-84440-C2-
2-P (A.T.), and No. PGC2018-093998-B-I00 (A.P.), and by
he Jun a de Ex emadu a (Spain) G an s No. IB16087 and
No. GR18079 (F.V.R., A.S., and M.L.C.), pa ially unded by
he ERDF. Use o compu ing acili ies om Ex emadu a Re-
sea ch Cen e o Ad anced Technologies (CETA-CIEMAT),
unded by he ERDF, is also acknowledged.
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[32] The e a e some o mal analogies be ween (T∗,θ)and
(T∗,θ), which may be su p ising a i s sigh . F om he
kine ic equa ion, he e olu ion equa ions o he ansla ional
and o a ional empe a u es, T and T , a e de i ed and hey
a e qui e di e en (see Re . [30] o hei explici exp essions).
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T, which is hei a e age, and o hei a io θ. The o mal
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[33] Since we a e in e es ed in cooling expe imen s, we ocus on he
egion o he (T∗,θ) plane in which <0.
[34] In ha case, he o a ional empe a u e anishes and 2only
depends on T∗, being p opo ional o √T∗(Ha ’s law [44]).
[35] No e ha 2is nega i e de ini e, so ha i s absolu e alue
dec eases wi h inc easing θ.
[36] No e ha , since bo h 2and 3a e p opo ional o √T∗,
θin
Eq. (7) is independen o T∗.
[37] No e ha 
θis bounded om abo e by i s maximum possible
alue 
θmax =2−3κ, o which i ends as α→1 o allβ.In
he pa icula case κ=2
5,weha e
θmax =4
5.
[38] Fo longe imes, e en ual c ossings a e unobse able since he
gas has almos eached he s eady s a e. The cu es in Fig. 3
ha e been ob ained by nume ically in eg a ing he nonlinea se
o equa ions (3) o e a su icien ly ine g id o ini ial condi ions.
[39] Wi h he only excep ion o he egion β≈0, in which we
know ha he angula ku osis and he ansla ional- o a ional
co ela ions a e mo e impo an [30].
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