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Entropy production at criticality in a nonequilibrium Potts model
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Ne w J . P hy s . 22 (2020) 093069 https://doi.o rg/10.1088/ 1367-2630/abb5f0
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P APER
Entrop y production at criticality in a nonequilibrium P otts
model
Thomas Mart ynec ∗ , S abine H L Klapp and Sar ah A M Loos
I nstitut e for Theo r etical Ph ysics, T echnisch e U ni v ersit ¨
at Berlin, H ar denber gstr . 36, D-10623, Berlin, German y
∗ A uthor to whom an y corresponde nc e shoul d be addressed.
E-ma il: mar [email protected]
Ke y w o r d s : nonequilib rium phase transitions, critical beh a vior, entro py pro duction , M on te-Car lo simulations
Abs t r a ct
U nder standing nonequilibr ium systems and the consequences of irre versibility f or the syst em’ s
beha vior as compar ed t o the equ ilibrium cas e, is a fundamental question in statistical physics.
H ere , we in ves tigate two t ypes of nonequilibr i um phase tr ansitions, a second-or der and an
infinite-order phase t ransition, in a prototy pical q -state vect or P otts model which is dr i ven out of
equilibr ium by coupling the spins to heat baths at tw o different t empera tures. W e discuss the
behavior of th e quantities th at are ty pically considered in the v icinity of (equilibr ium) phase
tr ansitions, lik e the specific heat, and moreo ver in vestigate th e beha vior of the entropy production
(EP) , wh ic h directly quan tifies the irre versibility of the process. F or the second-order phase
tr ansi tion, we sho w that the univ ersality class remai ns the same as in equilibrium. Fur the r , the
der i vative of the EP r ate w ith respect to the temper ature diverges with a power -la w at the cr itical
point, but displa ys a non-univ ersal critical exponent, which d epends on the t emperature
difference, i.e., the st rength of the dr iving . F or the infinite-order tr ansition, the der ivati ve of the EP
exhibits a maximum in the disordered ph ase, similar to th e specific h eat. H o wever , in contr ast t o
the specific heat, wh ose maximum is independent of the st rength of the dr i ving , the maximum of
the der i vative of the EP g ro ws with increasing temperature difference. W e also consider ent rop y
fluctuations a nd find tha t their skewness increa se s with th e dr iving st rength, in both cases, in the
v icinity of th e sec ond-order t ransition, as we ll as around the infinite-order tr ansition.
1. I ntr odu c tio n
Phase transit ions ar e ubiquitou s i n nat ur e and gene rally occur in equilibrium as w ell as nonequilibr ium
systems. In eithe r case, the transit ion is d ue to intern al interactions and often goes along w ith the breaking
of spatia l symmetries as a rea ction to the variation of a contr ol paramet er below its critical va lue, detecta ble
by the emergence of an appropriate or der parameter . Ph enomen a like the occurren ce of phase t ransitions in
one spatial dimension ar e solely obser v ed in nonequilibrium s yst ems [ 1 – 13 ]. In c ontras t, other pr operties
r elat ed to phase transiti ons are identical an d thus do not allow per c eiving whether a syst em is in a state of
thermal equ ilibrium or not. In equilibrium, it is well estab lished that continu ous ( second-o r der) phase
transition s ar e accompan ied b y po w er -la w div ergences of multiple measurable quantities, such as the
magnetic susceptibility or the spin – spin correlation length (for Is ing-lik e models) and there ar e alread y
n umer ous exam ples f or nonequilibrium syst ems that can be charac t erized in this manner as well [ 14 – 20 ].
H ow e ver , a general theor y for nonequilibrium p hase tr ansitions is stil l missing , and it is not per s e clear
whethe r the critic al exponent s o f a system stay the same (i.e., the system remains in the same universalit y
class) wh en driv en a way fr om equilibrium .
T o analyze nonequilibrium system s there exists ano ther , yet alm ost complet ely unconnected, t ool, that
is, the en tr opy pr oduction (EP). This quantity is stric tly posit i v e for nonequ ilibrium and exactly zero for
eq uilib rium syste ms. The total EP i s a fu ndament al q ua ntit y of g r eat import ance in stat istical physics that
alread y plays a central role in (stoc hastic) t hermodynamics and informat ion theor y , and it is k n own to
fulfill various law s, including the famous fluctuation theorem s [ 21 – 27 ]. M or eo ver , the EP can be defined in
© 2020 Th e A utho r(s). Pu blished b y IOP Pu blishin g Lt d on b ehalf of th e I nstitut e of Ph ysics and Deutsche Ph ysikalische Gesellsch aft

Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
a very general manner (not system- specific) and is, in principle, a meaningful quantit y for an y complex
system. This is b ecause it so lely de pends on (state and tr ansition) probab ilities, and d oes not rely o n
conc epts like ener g y or temperatur e. Thus, it can be defined and calculated a lso , e.g ., in non-ph ysical
systems, like social dynamics, or opinion format ion, which ar e at the same time known to underg o p hase
trans ition s [ 28 , 29 ]. F or thes e r eas ons it is tem pting t o in v estiga te this quantity with r egar d to critical
behavior in nonequ ilibrium systems.
A few recent studies ha ve alr e ad y started to in ves tigate the behavior of total EP around criticalit y in
differ ent lattice-based m odels by m eans of M onte-Car lo simulations a nd mean field theor y [ 30 , 31 ]. Fo r
example, for a one-dimen sional KPZ interfac e g r owth model [ 32 , 33 ] the rat e of EP ar ound the critical
point of a first-order phase transition was c alculated . Fur t he r mo re, t he re are s e v eral studies con sidering
spin systems wit h up – down ( Z 2 ) sy mmetr y . These inc lude an interacting lat tice gas model in contact w ith
two heat and particle reservoirs [ 30 ], the majorit y vote m odel [ 34 , 35 ] , and an Ising mod el externally dr i ven
by an det erministically oscillating magnetic field [ 36 ]. In all cases, t he EP rate was either found to jump or
displa y an infl ection point at criticality , s uch that its fi rs t deriva ti ve wi th r es pect to a n appr opriate contr ol
parameter exhibits marked behavior around the c ritical point. In partic ular , t he der i vativ e shows a
discont in uit y or a p o wer -la w d i vergence, b earing a resemb lance w ith the susce ptibil ity , the sp in – spin
corr elation len gth a nd the specifi c heat. Of special int er est for the presen t wor k is a study considering a
varian t of the s quar e latti ce Is ing m odel with nea r est- ne ighbor interac tions, w hose spins ar e in contact with
two heat baths at temperat ur es T 1 and T 2 in a checkerboar d arran gement [ 37 , 38 ]. In this study , the
deriva ti ve of the EP was foun d to div erge ar ound the second- or der ph ase transiti on with the sam e critic al
exponent as the spe cific heat [ 38 ]. Th is rais es the ques tion whether th ese di verging quantities generally
beha ve a like at criticality.
In the present paper , w e aim to generalize the pr evious findings to s pin symmetries differ ent from Z 2 ,
and to other ty pes of phas e trans itions . T o this end, w e cons ider a nonequilibri um q -s tat e vect or P otts
model aroun d criticalit y . Depending on the value of q , t he model displays either a secon d-order phase
transit ion from a paramag netic (PM) to a ferromagnet ic (FM) phase or an infinite-or der phase tr ansition
[ 39 – 41 ] fr om a PM to a quasi long-range or der ed Berezins kii – K ost erlitz – Thouless (BKT) phase. W e
in vestigate the model in t he v icinit y of both t y pes of pha se t ransitions u nder noneq uilibrium cond itions. In
particular , we couple the spins to hea t bat hs at two differ ent temperatures T 1 and T 2 in such a way , that all
nearest-neig hbors o f each spin are in cont act only with spins coupled to the respec tiv e o ther heat bat h.
H ence, tw o sublattices ar e formed in a checkerboar d arrangement similar to [ 37 , 38 ]. As a consequence of
the two in v olv ed tem peratures , a net heat flow fr om the hotter to the colder heat r eservoir is induced. This
setup dri ves the syst em in a nonequilibrium st eady stat e wher e it con stantly produc es entrop y , which is
export ed t o the e n vir onm ent. W e in ves tigat e the s ys t em n umerically using M on t e Carlo simulations with
Glauber dynamics. T o deep en our understanding of nonequ ilibrium phase transit ions, we study the EP and
its fluctu ations, as well as standard quant ities such as the magnet ization. U sing the finite-size sc aling
techniq ue [ 42 – 45 ] to carefully analyze the c ritical behavior based on numerical data, we dedic at e a detailed
analysis to the q uestion whether the specific heat and t he derivativ e of the EP b eha v e alike at cr iticality for
both ty pes of phas e transition s.
Our an alys is r eveals that the deri vativ e o f the EP ra te wi th r espect t o tem peratur e in the P M disor der ed
phas e of the four -stat e vect or P otts model sha r es indeed similari ties with the specific heat. Specifically , it
sho ws pow er- la w beha vior as function of the dista nce fr om criticali ty , ho w ever , in contras t to the specific
heat, wit h non-uni v ersal scaling exponent. Additiona lly , its maximum value diverges at critic ality and also
shows power -la w be ha vi or as funct ion of system size w ith a non-u ni versal sc aling exp onent. In contr ast, for
the X Y model with q →∞ , t he EP rate resembles t he behavior of th e specific heat. Both q uan tities do not
d i v e r g ei nt h ev i c i n i t yo ft h et r a n s i t i o nf r o m t h eP M t ot h eB K Tp h a s e .
2. M odeling and simulation details
2.1. The q -state vector Potts mode l
The Hamiltonian of the q -stat e vect or P otts model (or the q -s tate clock model) with neares t-neighbor spin
interactions on a discr ete lattice without an y e xternally applied magnetic field is defined as
H = − J 
 ij 
σ i · σ j = − J 
 ij 
cos  θ i − θ j  . (1)
He re, J repr esen ts the couplin g c ons tant betw een in teracting spin s which is set to un ity and ther eby
fav ors fer r omagne tic order . The sum i n equ ation ( 1 )r u n so v e ra l ln e i g h b o r i n gl a t t i c es i t e s  ij  .W ec o n s i d e r
squar e lattices in two dimens ions with a total n umber of L 2 spins, where L the lateral extension o f the
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Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
Figure 1. Illu str ation of the q -state v ect or P otts model on a tw o- dimen siona l squar e lattic e with periodic boun dary co nd itions.
The spins are coupled to heat baths at two diff er ent temperatures T 1 and T 2 , her e indicated b y the colors red and blue,
r espectiv ely , w ith a ch eckerbo ar d arrangement. Black lines r epr esent the near est-neighbor int eractions of str ength J ,w h i l er e d
and blue lin es highligh t the tw o sublattic es formed b y the co upling t o heat baths of differ ent t emperatur es.
system which we refer to as t he ‘system size’ . The spins σ i = e x cos( θ i ) + e y sin( θ i ) = [cos ( θ i ), sin ( θ i )] ar e
r epr esent ed as two-com ponent unit v ect ors in the x – y plane located on discr ete and equidistant positions i
on the lattic e. W ithin the q -sta te vector P otts model, the possible angles θ i ∈ [0, 2 π ]o f t h es p i n sa r e
gi v en b y
θ i = 2 π a
q ,( 2 )
where t he integers a = 0, 1, 2, ... , q − 1 deter mine the possibl e orient ations of the spins. At q = 2, the
model r educes to the classical I sing model with up – down ( Z 2 ) spin sy mmetr y , whereas in t he limiting case
q →∞ , the model corr esponds to the XY model wher e the spin orient ations ar e continuous within the
plane. In what follows , we focus on the cases q = 4a n d q →∞ .F o r , q = 4 the model (i.e., t he
Ashkin – T eller model) shows a sec ond-or der phase transition from a PM to a FM phase similar to the Ising
model, yet w ith different charac t erist ics, i.e., different critic al e xponents [ 46 ]. I n c ontra st, in th e
two-dimensional XY model (where q →∞ ), ther e e xists no long-range or der ed FM phas e at finit e
temperatur es as stated by the Merm in – W ag ner theor em [ 47 ]. Ins tead, the system undergoes an
infinit e-or der BKT transition fr om a PM to a BKT phas e.
In or der to s tudy the d ynamical e v olution of the system in pr esence of thermal noise, w e perform
M onte-Carlo simu lations w ith single sp in-flip Glauber dynamics. Here, the rate w μν ( i ) for a tr ansition of a
randomly chosen s pin σ i from state ν (before th e spin flip) to μ (after the spin flip) depends only on the
tempe rat ure T k of the heat bath the consider ed s pin is coupled t o , as well as on t he en erg y differ ence
Δ H = H ( ν ) −H ( μ ) r elated to a flip of spin σ i ,t h a ti s ,
w i
μν = 1
2  1 − σ i tan h  Δ H / T k  . (3)
If al l sp ins σ i are expos ed to a single heat bath at t emperatur e T , the system (that is initially prepar ed in
a configuration w ith random spin orientations) eventually reac hes a st at e of thermal equilibr ium and t hus,
does not pr oduce entr op y , Π= 0( s e e s e c t i o n 3 for a definition of E P). I n contrast, here w e driv e the system
into a nonequilibr ium s teady state by coupling the spins σ i to tw o differ ent heat ba ths T k ( k = 1, 2) which
a r ek e p ta tt e m p e r a t u r e s T 1 and T 2 . W ith this setup , the system is out of eq uilibrium whenever T 1  = T 2 .
There is a const ant heat flux ˙
Q fr om the hotter to the colder heat r eservoir that goes along w ith a constant
rat e of EP , Π > 0( s e es e c t i o n 3 ). Our setup splits the sys tem into tw o sublatti ces , L 1 and L 2 ,e a c h
containing all spins connected to the bath at T 1 or T 2 , r especti vely . All four neares t-neighbors of a spin σ i
a r ec o u p l e dt ot h er e s p e c t i v eo t h e rh e a tb a t h , y i e l d ing a ch eckerboard c onfiguration as illus trated in
figure 1 . In the following , we fix T 2 ,b u t v a r y T 1 and c alculate all o bservables as funct ion of the mean
tempe rat ure T = ( T 1 + T 2 ) / 2.
3 . E n t r o p yp r o d u c t i o ni nt h ev e c t o rP o t t sm o d e l
A ke y quantit y that distinguishes systems out of t hermal equilibr ium fr o m those in equilibr ium is the
constant net p r oduction of entrop y . In gene ral, the dyn amical e v olut ion of ph ysical syste ms that possess a
3

Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
finite set ν ∈ Ω of discret e micr ostates (e.g., the spin configurations of the vector P otts model at finite q )
can be des cribed as con tinuous time M ar k ov c hains . For such syst ems, the time- dependent system (or
Shannon) ent r op y [ 48 ] is given b y the Boltzmann – Gibbs expres sion
S ( t ) = − 
ν
p ν ( t )l n  p ν ( t )  . (4)
H ere, the sum runs ov er all micr ostates (i.e., spin configurations) and p ν ( t ) repr esents the occupation
pr obabilit y of stat e ν at t ime t . By coupling the system t o an infinitely l arge heat bath at temper atur e T (th at
alwa ys maintains a st at e of ther mal equilibr ium), we can for mulat e an expressi on for the time-dependent
change of entr opy [ 24 ] t hat one the o ne hand, o rig inates from t he total system inte rnal produ ction of
entr op y Π ( t ) and, on the ot her hand, by the ex change of entr opy Φ ( t ) with the envir onm ent,
∂ t S ( t ) =Π ( t ) − Φ ( t ) . (5)
In or der to formulat e explicit expr ess ions for Π ( t )a n d Φ ( t ), w e mak e us e of th e fact th at th e (genera lly
time-dependen t) occupation probabilities p ν ( t ) obe y a mast er equation
∂ t p ν ( t ) = 
μ  w νμ ( t ) p μ ( t ) − w μν ( t ) p ν ( t )  . (6)
The change ∂ t p ν ( t ) s tems first , from t he total incomi ng proba bilit y flow  μ  = ν w νμ ( t ) p μ ( t ) consisting of
all pos sible s tat e trans itions μ → ν happen ing with tra ns ition rat es w νμ ( t ). The second contributive t o
∂ t p ν ( t ) is the total outgoing pr obability flo w  μ w μν ( t ) p ν ( t ) due to trans itions ν → μ .
For syst ems in ther mal equ ilibrium, the detailed balance (DB ) condition, w νμ p μ = w μν p ν , holds for all
μ and ν . W hen DB is violated, there ar e non-vanishing loc al probability flow s bet w een certain microstates,
i.e., w νμ p μ − w μν p ν  = 0. As a consequen ce, the sys tem cons tantly pr oduces entr opy Π ( t ) > 0, which is
gi v en b y [ 49 ]
Π ( t ) = 1
2 
μ , ν  w νμ ( t ) p μ ( t ) − w μν ( t ) p ν ( t )  ln w νμ ( t ) p μ ( t )
w μν ( t ) p ν ( t ) . (7)
Equation ( 7 ) obe ys the thermodynam ically expected properties: Π ( t ) nullifi es in t hermal equilibrium, and is
strictly positi v e otherwis e, in ac cor danc e with th e second law of ther m odynam ics. For nonequilibrium
stationary states (no time-dependency), Π ( t ) =Π ,o n eh a s ∂ t S = 0 and conseque ntly fr om eq uation ( 5 ),
Π=Φ . Additionally , equation ( 7 ) reduces t o
Φ=Π= 
μ , ν
w νμ p μ ln w νμ
w μν
= 
μ , ν
j νμ ln w νμ
w μν
. (8)
Equation ( 8 ) can be computed n umerically by av erag ing over man y tr ansitions μ → ν from t he cur r ent
state μ of the Mar k o v chain (i.e., the curren t spin c onfigura tion of the lattice ) in the st ead y state. In spin
systems w ith discrete spin or ientations like the vector P o tts model wit h finit e q , state trans ition s μ → ν
corr espon d to the flipping of a r andom ly chos en spin σ i on lattice site i . Thus, the sum in equation ( 8 )c a n
b ew r i t t e na sa na v e r a g e o v e ra l ll a t t i c e s i t e s
Φ=Π= 
i 
ν  w i
νμ ln w i
νμ
w i
μν  . (9)
He re, w i
νμ [see equation ( 3 )] corr espon ds to t he G lauber - t ype flipping rate of the spin σ i on lattice s ite i
(which is co nnected to the he at bat h at T k ), inducin g a transiti on fr om the curr ent sta te μ to st ate ν due to a
change of the cur r e nt orientation θ i of spin σ i to an y other allow ed one. The stead y ex ch ange of entrop y
with th e en v ir onment r esults fr om the ne t heat fl ux ˙
Q fr om the hotter to the colder s ublattice. W e here
employ the sign con vention ˙
Q > 0 for the heat flow fr om hot to cold. Due to energ y conservation (and
because no external fields, for ces or further g radients act on the system), all of the three r elevant heat flows
transport the same amount of energ y per timest ep: the fl ow fr om the h otter heat bath T 1 ( h e r ew ea s s u m e
for a mome nt T 1 > T 2 ) to the corres ponding sublattice L 1 ,t h e fl o wf r o m L 1 to L 2 ,a n d t h eh e a tfl o wf r o m
L 2 to the cold bath at T 2 (or e v er ything reversed, if T 2 > T 1 ). This am ounts to a n o verall entropy flow to
the en vironment of Φ= | ( ˙
Q / T 2 ) − ( ˙
Q / T 1 ) | = ˙
Q | T 2 − T 1 | / ( T 1 T 2 ).
Equation ( 9 ) can be used to c alculate the mean of the EP rate, Π , for systems wit h a finite set Ω of
discr et e micros tat es (i.e., the vect or P otts model with q = 4) by means of Mont e-Carlo simulations. In
sect ion 5 , we sho w r esults for the EP rate per spin, which is given by π =Π / L 2 .
A problem with equation ( 9 )i st h a ti tc a n n o tb eu s e dt oc a l c u l a t e Π in s yst ems with con tin uous degrees
o ff r e e d o m ,a si ti st h e c a s ef o rt h ev e c t o rP o t t sm o d e lw i t h q →∞ ( X Y model). As an alternative, w e
4

Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
calculate the EP rate by following indi vidual stochastic tra ject ories consisting of cons ecuti vely e x ecuted state
trans ition s s n − 1 → s n betw een micr os tates as the syst em dynam ica lly evolv es via the r eorientation of sing le
spins σ i [ 24 ]. Suc h trajectories corr espon d to a sequence ( s 0 → s 1 → s 2 ... → s n − 1 → s n ... → s l − 1 → s l )
consisting of l st ate transit ions, e ach conne cted with a tr ansition rate, w i
s n s n − 1 , g i v en by equation ( 3 ). N ot e
that ea ch indi vidual stat e that is part of the traject or y simply c orresponds to one of the micr ostat es of the
system, s n ∈ Ω , i.e., to one of the possible s pin configuratio ns. Since there ar e infinitely man y states for
q →∞ , we make use of t he fact t hat each tr ansition s n − 1 → s n [for whic h we know the trans ition rate
w i
s n s n − 1 acc ording to eq uation ( 3 )] along the stochas tic path due to a random r eorientation of a spin σ i is
associ ated w ith a (sto chastic) e ntr opy exchange w ith t he en vironment Δ φ = ln  w i
s n s n − 1 /w i
s n − 1 s n  [ 24 ]a n d
an associated local heat ex change of Δ φ/ T k . As a consequenc e, the total c hange of entrop y along an
indi vidual stochastic p ath c onsis ting of l transiti ons r eads
Δ φ ( l ) =
l

n = 1
ln w i
s n s n − 1
w i
s n − 1 s n
. (10)
In the limit of infinitely long t rajectories, l →∞ ,e q u a t i o n( 10 ) divided by the length l of the t raject ory
become s identic al to the ense mble av er aged medium E P rate Φ due to the ergodic ity of t he syst em. In a
steady state t his is fur ther identical t o the t otal EP rate as calculated ac cor ding to equation ( 9 )
lim
l →∞
Δ φ ( l )
l =Φ=Π . (11)
In addition, we also use equation ( 10 ) to obtain dist ributions P [ Δ φ ( l = 100)] for the four -state model
as well as the version wit h q →∞ (see section 4 for de tails).
4. M easur ement details and parameter sett ings
In th e pres ent s tudy , sim ulations of the v ect or P otts mo del with neare st-n eighbor int eractions a r e
performed on square l attices w ith lateral extension r anging from L = 16 to L = 96. T his means that we
consid er L 2 = 256 up to L 2 = 9216 s pins. Befor e calculating an y ph ysical quantit y , we firs t let the syst em
ev o l v e f o r 5 × 10 4 Mont e Carlo s teps (MCS) [where one MCS consists of L 2 spin flip attempts with
spin – flip rates w i
μν accor ding to e quation ( 3 )] to assur e that the system has reac hed a steady state. W e then
let each system further e v olv e up to a maxim um of 10 6 MCS and use ( depen ding on the system siz e L )
between 100 and 1000 realizations for each parameter setting (i.e., combination of T 1 and T 2 )i no r d e rt o
guarantee the con vergenc e of av erage quan tities.
In or der to quantify the phas e behavior of the system, we calculate the magnetic order parameter
m = 1
L 2 


  
i
cos θ i  2
+  
i
sin θ i  2
. (12)
The value of m ∈ [0, 1] i s a measur e for the spin ordering in the syst em. For perfect order , m = 1, while
in a completely dis or der ed syst em, m = 0. W e also define the magnet ic or der parameters for t he tw o
sublatti ces L 1 and L 2
m k = 1
2 L 2 



 ⎛
⎝ 
i ∈L k
cos θ i ⎞
⎠
2
+ ⎛
⎝ 
i ∈L k
sin θ i ⎞
⎠
2
, (13)
where t he index k = 1, 2 den otes the r especti ve sublattic e.
T o precisely determ ine the val ue of the critical t emperatur e T c where t he phase transit ion (from the PM
to FM or to the BKT phas e) sets in, we comput e the f ourth-or der Binder cumulan t [ 42 ]o ft h e m a g n e t i c
or der parameter m
U 4 = 1 −  m 4 
3  m 2  2 , (14)
which is universal at c riticalit y . The crit ical value T c of the c ontr ol paramet er is giv en by the inters ection
point of U 4 for d iffer ent l ateral sizes L of the system.
In addition, we calculate the specific heat per lattice site whic h is giv en by
C v = 1
T 2   E 2 − E  2  , (15)
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w h a tc a na l s ob ee x p r e s s e sa sd  E  / d T . H er e,  E  corr espon ds to the average ener g y per spin. The specific
heat is known to exhibit pow er -law scaling as t he critical temperature T c is approac hed from the PM
disor der ed phase. Addition ally , C v peaks at T c where its m aximum s hows po w er -la w scaling as function of L
whic h is often uni v ersa l [ 50 ].
The a verage EP rate per spin, π =Π / L 2 , is c alculated accor ding to eq uation ( 10 ) for the four -state
vect or P otts model, while equation ( 11 ) is used in the c ase of the XY mo del ( q →∞ )w h e r et h es p i n
orientation is contin uous. Ho w ever , π can also be ob tained from e quation ( 11 ) for the four -stat e model. In
both cases , we check wheth er the chan ge of π , with r espect t o the c ontr ol paramet er T , d π/ d T ,s h o w s
uni vers al f eatures (s imilar to the specific heat C v ) r egar ding its scaling behavior as function of system siz e L
ar ound the critical poin t T c of the phase tran sition.
Distri butions P ( φ ) of the c hange of entr op y φ =Δ φ ( l ) for trajectories of length l = 100 are obtained v ia
equation ( 10 )f o r q = 4a n d q →∞ .W ec a l c u l a t e φ for t he w hole lattice and the individual sublattices L 1
and L 2 , respectiv ely . This is done by defining tr ajectories that only account for the change o f entr opy
induced by state tr ansitions due to a reorientation of sp i n sw h i c h a r ec o n n e c t e dt ot h er e s p e c t i v eh e a tb a t h
T k ( k = 1, 2). The distributions P ( φ ) ar e obtained fr om at leas t 10 7 trajectories.
5. R esu lts
In this section, w e presen t a numerical in ves tigation of the nonequilibr ium v ector P ott s model wit h discret e
( q = 4) and continuous ( q →∞ ) symmetry . In both cas es, w e find that t he nonequilibr ium model exhibits
the same ty pe of phas e transiti on as in th e equili brium case. W e ther efor e focus for q = 4 on the trans ition
fr om the spin -disor der ed PM t o the spin-or der ed FM phas e, wher eas for q →∞ w e analyze the BK T -like
transition fr om the disorder e d PM to the quasi long-r a n ge order ed BKT phase. In both cases the t ran sitions
are contin uous in t he or der par amet er . T o characterize t he critical b eha vior , we stu d y the specific heat and
the EP , and we compar e the res ults in the v ici nity of the cr itical point for both kinds of phase transition .
5.1. Phase t ransition in the discret e vect or P otts model w ith q = 4
Before w e numeric ally in vestigate the phase tr ansitio n of our nonequilibrium m odel, let us briefly r ev iew
some important properties of the equilibrium ver sion of the four -state vect or P otts model. It is well known
[ 51 – 53 ] t hat the equilibrium model exhibits a second-or der phase transition at T eq
c = 1 . 13, which is half
the critical tem peratur e of th e cla ss ical I sin g model ( T eq
c = 2 . 26) that exh ibits up – down symmetry . The
cr itical exponents of the four -state version are different fr om the Ising model [ 46 ].
T o beg in with t he analys is of the nonequilibr ium model, we consider the behavior of the
ense mble-a v eraged m agnetizati on m [see equation ( 12 )], which ser ves as a g lobal or der parameter . F igur e 2
displa ys m as function of th e mean t emperatur e T for f our differ ent (fix ed) values of T 2 and var ious syst em
sizes rang ing from L = 16 t o L = 96. The most pr ominent observation is that w hile decreasing the m ean
temperatur e from high values, the order parameter inc reases and eventually appr oac hes its maximum value,
m = 1 (reflecti ng perfect spin or der). Th is implies the e xist enc e of a s table FM phase at lo w bath
temperatur es, althoug h the sys t em is clearly out of equilibrium.
Figur e 2 (a) i ndicat es th at the nonequilibri um phase tran siti on occur s at a tem peratur e wh ich is
comparable to the one of the equilibr ium model, T eq
c = 1 . 13. Ho w ever , a closer inspection reveals that the
pr ecise value of T c depends on the fix ed tem peratur e T 2 in such a way that T c b ecomes small er as T 2 is
shifted awa y from the c ritical value T eq
c of the equilibrium model. This cons picuousnes s is f urther
confir med by figu r e 2 (b), wher e (for L = 24 and L = 64) m is plotted as function of T fo r different val ues of
T 2 . R emarkably , the shift of the temperatur e r egion (com par ed to the equilibrium model) wher e m as
function of T incr eases t o large values (indicating the emergenc e of spin or der) i s found to be equally large
for bo th system sizes L . This, in turn, sig nals that the temperature shift of the cur ves is not a finite s ize effect
(whic h w ould van ish for L →∞ ), but an actual property of this n onequilibrium v ector P otts m odel. A
further inter esting observation from figur e 2 (b) is t hat the nonequilibrium vect or P otts model studied here
displays an or der ed phase, e ven if one of the heat b ath’s temperatures is higher t han the critical temper atur e
T eq
c of the corr esponding equilibrium model. H ow ever , the critical mean temperature is alwa ys belo w the
equilibrium value, as we w ill discuss below.
T o precisely analyz e the dependency of T c on T 2 (and T 1 ), we comp ute the Binde r cumulant U 4 as
function of T for different v alues o f L [s ee equation ( 14 )a n d b e l o w ] .F i g u r e 3 (a) sho ws the cro ss ing of the
r especti ve lines for two ex emplary temperatures T 2 , clearly confirming the aforementioned shift of T c .
Sinc e th e spi ns σ i are c oupled t o differ e n t heat baths, one m ight expect differ ences in the phase behavior
of the two sublattices L 1 and L 2 . Ho w ever , as one can s ee in figur e 3 (b), the Binder cumulant inters ects at
the sa me temperatur e T c in both sublatti ces . Th is sho ws that the trans ition fr om the P M to the FM phase
occurs collectiv ely in the entire sy stem at the same temperatur e T c .
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Figure 2. (a) The ensemble-a v erage d magnetiza tion m vs mean temperature T = ( T 1 + T 2 ) / 2 in the v ecto r P otts model with
q = 4, for system sizes ranging fr om L = 16 to L = 96 (differ ent color s and sy mbo ls) and fix ed value for the temperature T 2 in
each panel. (b) The magnetization m in the model with q = 4a sf u n c t i o no f T for L = 24 and L = 64 and different v alues of T 2
rangin g fr om T 2 = 0 . 3t o T 2 = 1 . 5. The s olid g ray li nes cor r espond to m in the equilib rium mode l wher e T = T 1 = T 2 .T h e
dashed v ertical lines in both panels indicate the critical temperature T eq
c = 1 . 13 of the equilibrium mode l.
Figure 3. (a) Binder cumulant U 4 in the ve cto r P otts model w ith q = 4 and system siz es from L = 16 to L = 96. T he
int erse ction po int mar ks the va lue of th e critical temp eratur e T c .A t T 2 = 0 . 3w efi n d T c = 0 . 997(4), wh ile at T 2 = 0 . 5t h e l i n e s
inte rsect at T c = 1 . 075(8). ( b) Binder cumulant of the two sublattices L 1 and L 2 with q = 4f o r T 2 = 0 . 3a n d T 1 ranging fr om
T 1 = 1 . 5t o T 1 = 1 . 9f o r L = 24 to L = 48. The dashed lines mark the critical valu e T c where the Binder cumulant intersects. (c)
Phase diagram of the n on equilibrium v ector P otts model with q = 4 sho wing the bound ar y (circles) between the FM or dered
(indicated b y the blue shaded r eg ion ) and the PM disor dered phase as function of T 1 and T 2 . The critical temperat ur es ha v e been
obta ined fr om the cro ssing of the Binde r cumulan t for differ en t system siz es L .
For an overv iew o f t he critical temperatures in the plane spanned by T 1 and T 2 , we now look at the
nonequilibrium phase diagr am plotted in figur e 3 (c). The d iagonal (black solid) line where T 1 = T 2
corr esponds to the equilibrium model. For the nonequilibrium system ( T 1  = T 2 ), T c depends on T 1 and T 2
appr o ximately lin early in the vicinit y of equilibrium ( T 1 ≈ T 2 ) but the dependency becomes str ongly
nonlinear when T 1  T 2 or T 2  T 1 . This is clearly seen when one compar es t he actual phase boundar y
wit h the dashed cur v e corresponding to the line along which T = T eq
c holds, i.e., T 2 = 2 T eq
c − T 1 .O n ec a n
further see that when T 1 = T 2 (i.e., in t h e equilibrium model), the ph ase t ransition o c curs at the hig hest
mean temp eratu r e T eq
c .A ss o o n a st h e r ei sat e m p e r a t u r e d i f f e r e nc e between the two s ublattic es, th e
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Figure 4. (a) The specific heat C v vs T for T 2 = 0 . 3a n d T 2 = 0 . 5 and system sizes rangin g from L = 16 up to L = 96.
(b) P ower -la w scalin g of t h e specific heat C v vs the r educed temperatur e τ = | 1 − T / T c | in the diso r der ed phase fo r fix ed T 2
and system sizes ranging fr om L = 16 up to L = 96 indicat ed by differ ent colo rs and sy mbo ls. F or both values of T 2 the da she d
black line follo ws ∼ 2 / 3.
nonequilibr ium phase tr ansitions occu r at a lower critical temperatu r e, whic h de viates from T eq
c the mor e as
the d iffer ence Δ T = | T 1 − T 2 | between T 1 and T 2 incr eases. Mor eov er , there exists a new t y pe of cr itical
tempe rat ure T ∗
c = 1 . 700(2) with the follo wing pr operty : if one s ublattice has a temperatur e higher than T ∗
c ,
global order is des tr oy ed, irr especti v e of the temperatur e of the other sublattic e.
Ph ysically , one may understand the phase behavio r in the following manner . When heating the system
u pi nt h ep r e s e n c eo fat e m p e r a t u r ed i f f e r e n c e Δ T between the sublattic es, dis or der is fav or ed alr ead y at
low er s ystem -a veraged t emperatur es T , sho wing that a s maller amount of thermal noise destro ys the
long-range order . Consistent w ith our ph ysical intuit ion, a breaking of t he translational symmet ry (by the
temperatur e differ enc e between the sublattices ) r educes the stability of long-range or der . N ote t hat this is in
sharp contrast to the sit uation whe re an homog eneous external mag netic fie ld acts on t he system (breaking
the up – down s ymmetry) whi ch incr eases the sta bility of lon g-ra nge or derin g.
5.1.1. Critical behav i or of specific heat
After t he dete rminat ion of T c and its dependency on the tem peratur es of the two hea t baths, w e no w turn to
the in vesti gation of the thermody nam ic properties of our nonequilibrium spin model in the vic inity of the
phase transit ion. First, we calculate the specific heat C v [see equation (16)] as function of the m ean
tempe rat ure T for d iffer ent value s of L and T 2 . This quant ity is commonly considered in or der to
characterize second-order phase transit ions. As can be seen in figure 4 (a), C v peaks at a temperat ur e v ery
close to the values of T c that we hav e pr ev ious ly determined via the Binder cumulant (r ecall figur e 3 ).
For b oth d epi cted v alue s of T 2 , the precise location of t he peak depends on L in such a way that as t he
system size is increased, the temper ature wher e t he peak is located decr eases an d approac hes T c . W e suspect
that the peak is exa ctly at T c in the limit L →∞ , as it is w ell-known for the equilibrium v ersion of this
model. In ther mal eq uilibrium, the specific heat o f t he model is further kno wn to show univ ersal s caling
behavior with respect to the temperatur e, i.e.,
C ν ∼| 1 − T / T c | − α (16)
with α = 2 / 3 in the disorder ed phase [ 46 ]. I nter estingly , we find that the nonequilibr ium model also
displa ys a power -la w div ergence of C v . Mor eover , the critical exponent is the same as in equilibrium,
irr especti v e of the value of the tempera tur e g radien t Δ T = | T 2 − T 1 | among the two heat baths (as long as
T 1 and T 2 ⩽ T ∗
c ). This is e x emplarily illustrated for T 2 = 0 . 3a n d T 2 = 0 . 5i n fi g u r e 4 (b), wher e C v is
plotted for diff er ent system siz es (from L = 16 to L = 96) as function of the reduc ed temperatur e
τ = | 1 − T / T c | t ogether with straight (black das hed) lines (with slope − 2 / 3). W e che cked the scaling
behavior for various additional values of T 2 and all of them show a pow er -law scaling w ith α = 2 / 3,
demonstr ating the robustness of the critic al e xponent u nder noneq uilibrium conditions. T o analyze t he
crit ical beha vior based on our numerical data in d etail, w e employ the finite-size scaling technique. T o this
end, w e consider the positions of the peaks of C ν , wh ich giv e an appr o ximation f or the critical tem peratur e
as function of the system size L . For the equilibri um four -sta te vect or P otts model on a squar e lattic e, this
quant ity scales as ∼ L − ν , w ith the corresponding crit ical exponent ν = 2 / 3[ 54 ]. Als o for the
nonequilib rium model we ob tain ν = 2 / 3 for all values of T 2 , consist ent w ith the well-kno wn scaling la w
ν d = 2 − α [ 55 ]( w h e r e d = 2 is the spatial dimensi on of the lattice ).
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Figure 5. (a) The EP rate per spi n, π , as function of the mean temperatur e T for fix ed T 2 = 1 . 5 and system size L = 32 in the
model with q = 4. Th e solid black line c o rresp ond s to the equ ilibrium poin t wher e T 1 = T 2 and the dashed black line marks the
critical temperatur e T c of th e FM to PM phase tra nsition . (b) He atmap of the EP ra te p er spin, π , in the vect or P otts mode l w ith
q = 4 on a lattic e of size L = 32 for temp eratur es of the two sublattic es ran ging fr om T 1 = T 2 = 0 . 1u pt o T 1 = T 2 = 2 . 0.
5.1.2. Critical behavio r of total e ntropy product ion
Let us now c onsider the beha vior of the total EP which is a direct measur e for irrever sibility in the sense
that it quanti fies t he distance from equilibrium. T o star t wit h, w e find that the total EP per spin is al ways
positi v e, π> 0, wh enever T 1  = T 2 .M o r e o v e r , π is a con ve x f unction of the mean t emperature T wi th
minimum at t he eq uilibriu m point T = T 1 = T 2 ,w h e r e π = 0 [consistent wit h equ ation ( 7 )]. This can b e
seen in figur e 5 (a), which depicts π vs T for a n ex emplary settin g of T 2 = 1 . 5a n d L = 32 ar ound the
equilibrium mean temperatur e T = T 1 = T 2 = 1 . 5.
Dependin g on wheth er T 2 is higher or lower than T eq
c , the phase tr ansition of the noneq uilibrium model
lies below or abo ve that minimum (which is alwa ys located at T = T 1 = T 2 ). In other wor ds, if T 2 > T c ,
which is the sit uation consider ed in figur e 5 (a), the nonequilibrium ph ase trans ition occurs at the left- hand
side of t he minimum, whereas if T 2 < T c , the trans ition occurs at the right-ha nd side of it. Int er esti ngly , we
observe that π as function of T shows a b ump around T c = 1 . 11. In tha t sense, the function π ( T )i t s e l f
alr ead y signals the occur enc e of the phase trans ition.
The dependency of EP rate per spin on the temperatur es of t he two heat baths is plotted in figur e 5 (b),
which shows π for different co mbinations of T 1 and T 2 rang in g fr om 0.1 up to 2.0. As expected, π = 0
whenever T 1 = T 2 [i.e., no temperature gr adient Δ T is present and DB is fulfilled, see equation ( 7 )]. In
contrast to this, there is al ways a positiv e rate of EP ( π> 0) when T 1  = T 2 , consistent wit h the spec ial case
cons ider ed in figur e 5 (a). As the gradient Δ T increases, t he EP r ate increases roughl y π ∼ Δ T 2 ,n om a t t e r
whether t he system is in t he PM or t he FM phase.
T o r esolv e the EP along indi vidual stochas tic trajectories, w e plot distributions of the m edium EP P ( φ ).
Figur e 6 displa ys n umerical results f or P ( φ ) at the c ritical temperature T c [figur e 6 (a)] a nd abo ve T c
[figur e 6 (b)], f or traj ectories of length l = 100 (see section 4 for details). W e consider both, the distribution
of the entire syst em as a whole (top panels), and the se para te distributi ons obtained b y r estrictin g our
obse rvat ion to one of t he two subl attices, L 1 or L 2 , only (middle and bottom panels , res pectiv ely). For
example, the m iddle panels sho w the hist ograms of all detect ed values of the medium EP f r om spins that
belong to the subla ttice L 1 . Overall, t he main charac t eristic s seem to be q uite similar for the system at and
abov e the phase transition (compar e (a) and (b)). Let us now take a c los er look at the different
distributions. R emar kably , in both cases, P ( φ ) for the w hole lattic e exhibi ts a multi -peak ed structure [s ee
top p anels i n figu r e 6 ]. W hen in specti ng the corr es pondin g subla ttice distribution s, w e notice tha t the
mult i-peak st ruct ur e o f the whole system app ears to ar ise as a comb ination o f both sub lattice s. This is
r easonable, as the stochastic trajectories of the whole system expectan tly include both, man y contributions
fr om the hott er sublattic e (which flips mor e oft en), and s ome seldom c ontributions fr om t he c older
sublattice. I n fact, the multi-peaked s tructure look s like a c on v olution of the distributions fr om the
belonging su blattic es. Further , w e notice tha t P ( φ ) of the individual sublattic es have s mooth s ingle-peaked
sha pes. Furth ermor e, all di stribution s ar e discr ete, r eflectin g that the num ber of pos sible tran siti ons (and
thus, φ values ) is fin ite, becaus e of the discr et eness of the underlying spin dyn amics . For the colder
sublatti ce, L 2 , φ on ly takes a particularly small num ber of values. This is due to the fact that at low bath
temperatur es, the sublattice only explor es a small pa rt of the phase space, an d hence, the number of dis tinct
state transitions is small. For the hot t er sublattice, L 1 ,w en o t i c et h a tt h e m a x i m aa n dm e a nv a l u e s o f P ( φ )
lie at φ< 0 ( in both cases , (a) and (b)). This alone w ould v i olat e the second-la w of thermodynamics sinc e
it implies a negative mean EP rate. How ever , in its usual form, Φ=Π > 0, the second la w only applies to
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Figure 6. Dis tribu tion P ( φ ) of the s tochast ic mediu m entropy φ =Δ φ ( l ) t hat is produced i n t he s ystem a long indivi dual
sto chastic tra jecto ries of length l = 100 in the four -state v ector P otts model. The panels of (a) show P ( φ )i nt h ev i c i n i t yo ft h e
phase transition [wh ich is at T = T c = 0 . 997(4)] for a system with L = 64, at T 1 = 1 . 7a n d T 2 = 0 . 3. The top panel sho ws the
distributio n of the wh ole system. T he middle pa nel sho ws the distributio n o f the medium EP of th e spins wh ich be long t o the
subla ttice L 1 , w hile the on e at the b ott om sho ws th e distributio n for L 2 . (b ) Sho ws the c o rr espon din g distributio ns in th e PM
disor der ed phase, at T = 1 . 3 > T c , specifically at T 1 = 2 . 3a n d T 2 = 0 . 3.
the entir e syst em which cons ists of tw o sublattice s, an d the negati ve mea n value sim ply refl ects the heat
flo ws fr om the hotter t o the colder heat bath (ov erall, the entrop y is incr eased over time).
N e x t ,w es t u d yt h es y s t e ms i z ed e p e n d e n c yo ft h et o t a lE Pr a t e( p e r s p i n ) , π , ar ound t he critical point of
the phase transiti on. T o this end, w e consider a system where T 2 is fix ed to a value below T eq
c [see the left
panel of figur e 7 (a)], and anoth er one wher e T 2 > T eq
c [see the right panel of figure 7 (a) w hich is essentially
an enlarged ver sion of figur e 5 (a) for d ifferent L close to T c ]. Both parts of figur e 7 (a) indicat e that π is
ident ical for a ll system sizes for T values far a way from T c , i.e., all lines c ollapse on a s ingle c urves. I n
contrast , the lines split up around T c , t hus, around t he phase transit ion π de pends on t he system size L .
This resembles the behavior of the specifi c heat [recall figur e 4 ]. On e further observes the emergen ce of a
shoulder whic h gets more pr onounced while increas ing L . It i s, howe ve r , no te wor thy t ha t we do no t ob se r ve
the formation of a saddlepoint or e v en non-m onotonous beha vior for all consider ed sys tem sizes , i.e., until
the value L = 96.
In or der to stud y the behavior ar ound the cr itical point , w e inspect t he derivative of the EP rate d π/ d T
for various values of T 2 see fig ure 7 . I nt er estingly , d π/ d T peaks ar ound t he t emperatur e of t he phas e
trans ition . An ex ception is the case T 2 = T eq
c wher e t he total EP naturally vanishes and t hus does not peak.
M or eo ver , one observes a dependency of the maximum of d π/ d T on the value of T 2 which (for fixed L )
decr eases as T 2 appr oaches T eq
c .
T o analyze the nonequilibrium phase tr ansitions in more detail, w e perform a finite-size scaling , similar
to our in vestigation of the specific heat (see figur e 4 ). W e aim to stress t hat the application o f a finite-size
scaling analysis to t he EP at a nonequilibrium tr ansit ion is, t o our kno wledge, novel. Fir st, w e stud y the
scaling b eha vior of d π/ d T in the d is or der ed phase as function of the reduc ed temperatur e τ .S e c o n d ,w e
cons ider the peak heig ht as function of the sys tem size L . As can be seen in figur e 8 (a), d π/ d T sho ws
pow er -la w b eha vior ∼ τ ζ wi th an expo nent ζ , whos e precise value depends on t he dis tance fr om
equilibrium at the phase transition (i.e., on the value of Δ T = | T 2 − T 1 | ). Specifically , we detect pow er -la w
behavior of d π/ d T for a ll consider ed values of T 2 with a dec r easing value for ζ as T 2 approac hes T eq
c ,w h e r e
it nullifies. For T 2 = 0 . 3[ s e et h e l e f tp a n e li nfi g u r e 8 (a)] the expon ent r eads ζ = 0 . 175(11), while for
T 2 = 0 . 5[ s e et h er i g h tp a n e li nfi g u r e 8 (a )] ζ = 0 . 145(15) (see the dashed black lines). W hile the
pow er -la w b eha vior res embles that of the specific he at , ther e is a marked difference in the sense that t he
exponent ζ is no t constant ( such as t he exponent α of C v ), but depends on Δ T .I na d d i t i o n ,w ea n a l y z et h e
scaling b eha vior of t he maximum of d π/ d T as the system siz e L is inc r eased and show result s for T 2 = 0 . 3
and T 2 = 0 . 5i nfi g u r e 8 (b). A ccor ding to t he finite-siz e scaling theor y for equilibrium systems [ 42 ], a ll
div ergent quantities scale as ∼ L a /ν ,w h e r e a is the cr itical exponent of the pow er -la w decay of that ver y
quant ity . Thus, we test whet her the maximum of d π/ d T scales as ∼ L ζ/ ν ,w i t h ν = 2 / 3. Fr om our
numerical dat a, w e find d π/ d T max ∼ L 0 . 245 for T 2 = 0 . 3a n dd π/ d T max ∼ L 0 . 205 for T 2 = 0 . 5w h i c h i s
indeed in good ag r eement with ζ = 0 . 175(11) ( T 2 = 0 . 3) and ζ = 0 . 145(15) ( T 2 = 0 . 5) as obtained in
figure 8 (a). The fulfillment of the finite-s ize scaling relation shows indeed that the derivativ e of the EP rate
behav es as a div erg ing quantit y as the c ritical poin t of the phase t ran sition is approached. I t further
demons trat es that the finit e-s iz e scali ng theory is applicable to the EP rate, despit e the dependency of the
critical e xpone nt on the tem peratur e gradient between the tw o sublattic es.
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Figure 7. (a) The EP rate per spi n, π , as function of the mean temperatur e T for fix ed T 2 and system sizes ranging from L = 16
to L = 96. I n the left panel, the temperatur e of sublattice L 2 is fix ed to T 2 = 0 . 3 which is belo w the critical temperatur e T eq
c of
the equilib rium model, w hile in the right panel, th e temp eratur e of L 2 is T 2 = 1 . 5, which is abov e T eq
c [see also figur e 5 (a)]. Th e
black d ashed line s mark the critical tempe ratur e T c .( b ) D e r i v a t i v ed π/ d T of the EP rate as function of the mean temperature T
for differ ent fi x ed values of T 2 fr om T 2 = 0 . 3u pt o T 2 = 1 . 13 = T eq
c and system sizes ranging fro m L = 16 to L = 96. The black
dashed lines mark the cr itical temperatur e T c .
Figure 8. (a) P o wer -la w scaling of the der i vative of the EP rate as funct ion o f the reduced temperatu r e τ = | 1 − T / T c | fo r two
valu es of T 2 ( T 2 = 0 . 3a n d T 2 = 0 . 2) and system sizes ranging fro m L = 16 to L = 96. Th e black dashed line in the left panel
follo ws ∼− 0 . 175(11), while in the rig h t panel it follo ws ∼− 0 . 145(15). (b) M aximum of the der i vativ e of the EP rate as function
of system size L . The l eft panel shows t he sca ling of d π/ d T max at T 2 = 0 . 3 for system sizes fro m L = 16 up to L = 96. The black
dashed lines scales ∼ 0 . 245. I n the rig h t panel the same is plo tted for T 2 = 0 . 5 and the black dashed line follo ws ∼ 0 . 205.
5.2. BKT -like phase tr ansition in the continuous v ector P otts mo del with q →∞
N ow w e turn to the vector P otts model with q →∞ (also known as the XY model), wher e the spins can
fr eely r otate in the x – y plane, i.e., all spin orient ations σ i ∈ [0, 2 π ] are allow ed. As a consequence of the
continuou s spin symme try and the two-dimensio nal char acter o f the system, t her e exist s no long-r ange
order e d phase at finite temper atur es as stated by the Mermin – W ag ner theor em [ 47 ]. I ns tead, a quasi- long
range or der ed phas e, the BKT phas e, occurs at low bath t emperatures. W hile the infinite-or der transition
betw een the dis or der ed and the BKT phase is qui t e w ell underst ood in the equilibrium model [ 56 ],
none quilibrium BKT phase tra nsi tions ar e in gen eral les s under st ood. I n particula r , the question of ho w the
EP rate b eha ves at t his transition has, to the b est of our k nowledge, not been consider ed in earlier literatur e.
In the previous discussion of the case q = 4, w e hav e s een that the deriv ati ve of the t otal EP sh ow s critical
11

Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
Figure 9. (a) EP rate per spin π of the nonequi libri um v ector P otts mode l w ith q →∞ (XY model) as function of the mean
tem pera ture T for system sizes ranging fro m L = 16 to L = 64 w ith T 2 = 0 . 3a n d T 2 = 0 . 5. (b) H eatmap o f π in the X Y model
on a la ttice of si ze L = 3 2 for tempe ratur es of the two sublattice s rangin g from T 1 = T 2 = 0 . 1u pt o T 1 = 2 . 0a n d T 2 = 2 . 5.
(c ) De rivat iv e of t he EP ra te per s pin, d π/ d T in the XY m odel on a l atti c e of s ize L = 32.
behavior which partially res embles the behavior of th e specific heat. Let us now see if this analog y carr ies
ov er t o the BKT transiti on, whic h has very differ en t ov erall characteristics and, in particular , is not
accomp anied w ith a d ivergen ce of C ν at the critical tempera tur e whic h is g i v en by T eq
c = 0 . 892 880(6) [ 57 ]
in the equilibrium XY model [ 56 – 61 ]. In figur e 9 (a), we sho w res ults for π at T 2 = 0 . 3a n d T 2 = 0 . 5f o r
system sizes rang ing from L = 16 up to L = 64. As indicated there, the EP rate does not split w ith r espect to
L in the vicinit y of the phase transition. Instead, π is apparently size-indep endent in the d epicted
temperatur e range wh ich includes the BKT trans ition. In or der to visualize the EP rate for differ ent
combinations of T 1 and T 2 , we plot π in t h e T 1 – T 2 plane in figur e 9 (b) together w ith the derivativ e of the
EP rate with res pect to temperatur e, d π/ d T in figur e 9 (c) for syste m size L = 32.
A dditionally , C v for T 2 = 0 . 5a n dd π/ d T for T 2 = 0 . 3a n d T 2 = 0 . 5 are p l ot te d in fi g ure 10 .I nc o n t r a s t
to the PM t o FM transiti on of the f our -stat e vect or P otts m odel, C v in the nonequilibrium XY model does
not show an y feature like a divergence at criticalit y . In part icular , it only shows a peak ar ound T = 1 . 1, as
does the equilibrium XY model [ 58 ], whic h is abov e T c . Inter estingly , also the der i vati ve of the EP r at e w ith
r espect to t emperatur e, d π/ d T , does not peak in t he vicinit y of the c ritical point . Similar to the specific
heat, d π/ d T also show s the peak ar ound T = 1 . 1 which does not depend on L , i.e., the maximum of d π/ d T
does not div erge, but r e m ains constant for all cons ider ed syst em sizes. Ho w ever , we obser v e that the
maximum o f d π/ d T depends on the temperatur e diff eren ce | T 2 − T 1 | between the tw o subla ttices i n the
vicinit y of the peak as confirmed by c omparing figure 10 (b) with figur e 10 (c), wher e one observes tha t the
maximum val ue of d π/ d T at T 2 = 0 . 3i s l a r g e rc o m p a r e dt o T 2 = 0 . 5.
J ust as for the vect or P otts model with q = 4, we in vestigate the dist ribution P ( φ )o fe n t r o p y φ =Δ φ ( l )
that is pr oduced along stochas tic tr aject ories of length l = 100. T o this end, we plot P ( φ )f o ras y s t e m o fs i z e
L = 6 4i nt h eq u a s il o n g - r a n g eo r d e r e dB K T p h a s ea t T = 0 . 5w i t h T 1 = 0 . 7a n d T 2 = 0 . 3( i . e . , Δ T = 0 . 4)
in the top panel of figure 11 (a). The distribution for the whole syst em seems to be s ymmetr ic ar ound the
peak position of P ( φ ) w hich is located in the positiv e range, φ> 0 i n accordance w ith th e second law of
thermo dy nam ics . In con tras t, P ( φ )f o rs u b s y s t e m L 1 p eaks in t he negative r ange, and P ( φ )f o rs u b s y s t e m
L 2 peaks at a positiv e value of φ . T his differ enc e in the peak positions just reflects the expected en trop y fl ow
fr om the hot t o the c old r eservoir . Additionally , one observes dif fer ent skew directions for P ( φ )i nt h e t w o
subsyste ms. P ( φ )f o r s u b s y s t e m L 1 is slightly r ight-skew ed, while P ( φ )i n L 2 is a left-sk ew ed distribution.
T h i se f f e c tb e c o m e sm o r ep r o n o u n c e df o rt h es y s t e mi nt h eP Mp h a s e[ s e efi g u r e 11 (b)] w her e one clea rly
observes that P ( φ ) is skew ed in both sublattices. Since the distribution for L 2 is stronger s kew ed, the
dist ribut ion for t he whole system is also l eft-skewed.
12

Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
Figure 10. (a) Specifi c heat C v of t he nonequi libri um v ector P otts model wi th q →∞ (XY model) as function of the mean
tem pera ture T for T 2 = 0 . 5 an d system sizes rangin g fro m L = 16 to L = 64 . (b) S ho ws t he deriva tive, d π/ d T ,o ft h eE Pr a t ea s
function T fo r th e same system siz es an d T 2 = 0 . 3, while T 2 = 0 . 5i n ( c ) .
Figure 11. Dist ribut ion P ( φ )o ft h em e d i u me n t r o p y φ =Δ φ ( l ) t hat is produced in t he syste m along s tochast ic t raje ctorie s of
length l = 100 in the X Y model (wher e q →∞ ). The top pan el in (a) sho ws P ( φ ) belo w the critical po int in the B KT ph ase for a
system with L = 64 at T 1 = 0 . 7a n d T 2 = 0 . 3. The m iddle panel in (a) s ho ws P ( φ )f o r L 1 and the one at the bottom of (a) for
L 2 . (b) Sho ws the same in the PM diso r der ed phase with for T 1 = 1 . 9a n d T 2 = 0 . 3.
6. Conclusions and outl ook
In this paper , we hav e analyz ed the behavior of various crit ical quant ities and that o f the total EP r at e
ar ound the c ritical point in a nonequilibr ium q - sta te vect or P otts model (with q = 4a n d q →∞ ). The
nonequilibrium c haracter results fr om coupling the sp ins to two heat baths at differ ent temperatures. Based
on this non equilibri um m odel, we addr es s several ques tions : does the ty pe of pha se transiti on and the
crit ical e xponent s change by dr i vi ng the system a way fr om equilibr ium? D oes the EP e xhibit u ni versal
behavior ar ound a continuous phase transition? W hat h appens to the EP in the v icinity of an infinite-or der
phase transit ion?
Firs t, w e ha v e in ves tigat ed the model with q = 4 in the v icinity of the second- or der p hase transit ion. W e
found that the crit ical temperatur e of the transit ion de cr eases as the temperatur e differ ence betw een the two
heat baths increases. M or eov er , the behavior of the sp ecific heat res embles that of the equilibrium model,
i.e., it shows pow er -la w div ergenc e with c ritical expon ents that are independent of the temperatur e
differ ence. Int er estingly , the derivati ve of the EP rate with r espect to temperatur e behav es, to som e ext ent,
similar . It also sho ws power -la w diver gence. H owever , the value of t he scaling exponents does depend on the
temper ature difference and is thus non-uni versal. Concer ning the model w ith q →∞ , the specific heat as
well as t he derivative of the EP r ate do not show any noticeab le beha v ior ar ound t he infinite-order
transition fr om the PM to the quasi long-r ange or der ed BK T p hase. I nstead, both quantit ies hav e a finite
peak at a t emperatu r e ab o ve the c ritical temperature, i.e., in t he PM phase. As t he t emperature differ ence
between the heat baths incr eas es, the maximum va lue of the derivati ve of the EP r at e becomes mor e
pr onounced. I n total, our res ults pro vide e vidence that the der i vativ e of the EP behav es like a critical
quantity , but, as we r eport here, is n on-un iv ers al.
Finally , we aim at pointing out perspecti ves f or futur e work, starting with some questions directly
following fr om the pr esent work. F or the sake of g enerality one should stud y and compar e the beha vior of
13

Ne w J . P hy s . 22 (2020) 093069 TM a r t y n e c et al
the specific heat w ith the EP in o ther dimens ions and for differ ent lattice topologies. Further , althoug h the
BKT p hase tr ansition is not accompanied by a divergence of thermodynamic q uantities, in equilibrium it
still obe ys characteri stic sc aling dimensions [ 62 ]. A more detailed analys is of t his transition in the
nonequilibrium model, and, specifically , w ith respe ct t o the d er iva tive o f the E P ra te, re pre se nts a n
in te res ti ng ob j ec t ive of fu t ure re se arc h. F rom a t he ore t ic al p oi nt o f v i ew , i t wo ul d mo reo ver b e wo rt h to
think about the c onnection betw een EP and specific heat, which seem to beha v e analogously ar ound
crit icality , on a fundamental le vel .
Furthermor e, an inter esting nov el perspecti v e on the nonequilibrium model consi der ed here is the
r einterpr etation as a model with non-r ecipr ocal couplin g betw een interactin g isotherma l spins . T o be more
specifi c, a v ecto r -P otts m odel wher e int eracti ng spi ns ar e c oupled a mong e ach oth er with two d isti nct
coupling c onstants ( J 1 = J / T 1 and J 2 = J / T 2 ) and unif orm temperature f ollows the exact same equations of
motion s as our model (with two t emperatur es and identical c oupling c ons tants J ). This pro vides a
c o n n e c t i o nt os p i nm o d e l so nd i r e c t e dg r a p h s[ 63 – 69 ] ,a n dt ot h et o p i co fn o n - r e c i p r o c a li n t e r a c t i o n s ,
which is cur r ently a focus in noneq uilibrium statistical mechanics [ 70 – 72 ]. It w ould be inter es ting to
compare t he thermo d ynamic proper ties o f spin syste ms sub jected to d iffer ent d riving mechanisms, e .g.,
non-r eciprocal c ouplings, tem perature gradients, ext ernal fields and color ed noise.
Ac k n o w l e d g m e n t s
This work was funded by the Deutsche For sc hungsgem einsc haft (DFG, German Res ear ch
Foun dation) — Pr ojektn ummer 163436311-SFB 910.
OR CID iDs
Thomas Mart ynec https://or cid.org/0000- 0001- 5736- 6108
Sarah A M Loos https://or cid.org/0000- 0002- 5946- 5684
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