The hidden fluctuation-dissipation theorem for growth
eepl draft
The hidden fluctuation-dissipation theorem for growth
M´arcio S. Gomes-Filho and Fernando A. Oliveira Instituto de F´ısica, Universidade de Bras´ılia, Bras´ılia-DF, Brazil Instituto de F´ısica, Universidade Federal da Bahia, Campus Universit´ario da Federa¸c˜ao, Rua Bar˜ao de Jeremoabos/n, 40170-115, Salvador-BA, Brazil
PACS – stochastic analysis methods
PACS – Brownian motion
PACS – Noise
Abstract –In a stochastic process, where noise is always present, the fluctuation-dissipationtheorem (FDT) becomes one of the most important tools in statistical mechanics and, consequently,it appears everywhere. Its major utility is to provide a simple response to study certain processesin solids and fluids. However, in many situations we are not talking about a FDT, but about thenoise intensity. For example, noise has enormous importance in diffusion and growth phenomena.Although we have an explicit FDT for diffusion phenomena, we do not have one for growthprocesses where we have a noise intensity. We show that there is a hidden FDT for the growthphenomenon, similar to the diffusive one. Moreover, we show that growth with correlated noisepresents as well a similar form of FDT. We also call attention to the hierarchy within the theoremsof statistical mechanics and how this explains the violation of the FDT in some phenomena.
Introduction.—
The era of explicit stochastic process inphysics started with Langevin’s analysis of the Brownianmotion [1–8], by considering the equation of motion for aparticle moving in a fluid as [5]: m dv ( t ) dt = − mγv ( t ) + f ( t ) , (1)where m is the mass of the particle and γ is the fric-tion. The ingenious and elegant proposal was to modulatethe complex interactions between particles, considering allinteractions as two main forces. The first contribution rep-resents a frictional force, − mγv , where the characteristictime scale is τ = γ − while the second contribution comesfrom a stochastic force, f ( t ), with time scale ∆ t (cid:28) τ ,which is related with the random collisions between theparticle and the fluid molecules.The fluctuating force f ( t ), in eq. (1), obeys the followingconditions:( i ) the mean force due to the random collisions on theparticle is zero, (cid:104) f ( t ) (cid:105) = 0 , (2)( ii ) there is no correlation between the initial particlevelocity and the random force, (cid:104) f ( t ) v (0) (cid:105) = 0 (3) and ( iii ) the fluctuating forces at different times t and t (cid:48) are uncorrelated (cid:104) f ( t ) f ( t (cid:48) ) (cid:105) = 2 ρδ ( t − t (cid:48) ) , (4)where ρ is the noise intensity. A solution for (1) is [8] v ( t ) = v (0) + 1 m (cid:90) t f ( s ) exp[ − γ ( t − s )] ds, (5)which gives < v ( t ) > = < v (0) > exp( − γt ) and (cid:104) v ( t ) (cid:105) = (cid:104) v (0) (cid:105) exp( − γt ) + ρm γ [1 − exp( − γt )] . (6)In order to get the above relation we have used theconditions ( i ), ( ii ) and ( iii ). Now, considering that (cid:104) v ( t → ∞ ) (cid:105) = (cid:104) v (cid:105) eq as t → ∞ , we get for the FDT: (cid:104) f ( t ) f ( t (cid:48) ) (cid:105) = 2 m γ (cid:104) v (cid:105) eq δ ( t − t (cid:48) ) . (7)Equation (7) is the FDT in its simplest form, i.e. a relationbetween the fluctuating force f ( t ) and the dissipation γ .Note that the equipartition theorem states that (cid:104) v (cid:105) eq = k B T /m , where k B is the Boltzmann constant and T theabsolute temperature. It is noteworthy that the artificialseparation between the stochastic force and the dissipativeforce in the eq. (1) now disappears. An important relationp-1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b omes-Filho et al. was pointed out between them. This main relation is theFDT. We show below that there is a similar relation ingrowth phenomena. The growth fluctuation-dissipation theorem.—
Sincegrowth phenomena are widely found in many dynamicalprocesses in biology, chemistry and physics, the surfacedynamics has become an intense research topic in the lastdecades [9–12]. Although these systems are complex innature, they can be modeled in simple ways. For instance,the statistical character of the evolutionary process leadsus to an inhomogeneous surface, which can be describedby a height h ( (cid:126)x, t ), being (cid:126)x the position in d dimensionalspace and t the time. Two quantities play an importantrole in growth, the average height, (cid:104) h (cid:105) , and the standarddeviation w ( t ) = (cid:2) (cid:104) h ( t ) (cid:105) − (cid:104) h ( t ) (cid:105) (cid:3) / , (8)which is named as roughness or the surface width. Theaverage here is obtained over the sites. It should be men-tioned that, again, the fluctuation is the main physicalquantity, and we would say that, in statistical physics, it issecond only to the diffusion dispersion, eq. (6), since manyimportant phenomena and processes have been associatedwith it.The general characteristics of the growth dynamics wereobserved through some analytical, experimental and com-putational results [10]. For many growth processes, theroughness, w ( t ), increases with time until reaches a satu-rated roughness w s , i.e., w ( t → ∞ ) = w s . We can summa-rize the time evolution of all regions as following: w ( t, L ) = (cid:40) ct β , if t << t × w s ∝ L α , if t >> t × , (9)with t × ∝ L z . The dynamical exponents satisfy the generalscaling relation: z = αβ . (10)Different methods have been proposed to understandthis rich phenomenon. Here, we will focus only on pro-cesses that saturate. For example, the attempt to de-scribe the height evolution leads us to some dominanttype of Langevin’s equations such as Edwards-Wilkinsonequation (EW) [11]: ∂h ( (cid:126)x, t ) ∂t = ν ∇ h ( (cid:126)x, t ) + ξ ( (cid:126)x, t ) , (11)and the Kardar-Parisi-Zhang (KPZ) equation [12]: ∂h ( (cid:126)x, t ) ∂t = ν ∇ h ( (cid:126)x, t ) + λ (cid:126) ∇ h ( (cid:126)x, t )] + ξ ( (cid:126)x, t ) , (12)where the parameters ν (surface tension) and λ are relatedto the Laplacian smoothing and the tilt mechanism, re-spectively. The stochastic process is characterized by thenoise, ξ ( (cid:126)x, t ), which is defined as a simple form, a whitenoise: (cid:68) ξ ( (cid:126)x, t ) ξ ( (cid:126)x (cid:48) , t (cid:48) ) (cid:69) = 2 Dδ d ( (cid:126)x − (cid:126)x (cid:48) ) δ ( t − t (cid:48) ) , (13)where D is here the noise intensity. For the KPZ univer-sality class, we have the Galilean invariance [12]: α + z = 2 . (14)In this way, the KPZ equation, eq. (12), is a general non-linear stochastic differential equation, which can character-ize the growth dynamics of many different systems [13–22].Despite all effort, finding an analytical solution of the KPZequation (12) is not an easy task [23–28] and we are stillfar from a satisfactory theory for the KPZ equation, whichmakes it one of the most difficult problems in modern math-ematical physics [29–37], and probably one of the mostimportant problem in nonequilibrium statistical physics.It should be noted that there is a gap in our un-derstanding of what the FDT for the growth processis, since the intensity D is not directly connected withother properties of the force as in the Langevin equation(LE). To overcome this, let us remember that in the LEthe equilibrium is reached when we apply the equipar-tition theorem, eq. (6), which establishes that we have (cid:104) v ( t → ∞ ) (cid:105) ≡ v s = k B T /m , where v s is the final squaredaverage velocity or the “saturated” velocity, that leads toeq. (7).For the EW and KPZ equation we have as well thatthe noise increases h and w ( t ) linearly with time, and theterm ν ∇ h ( (cid:126)x, t ), which is always opposite to roughnessdecreases it. For 1 + 1 dimensions we have the exact results w s = (cid:114) cDν L, (15)being c = π for Edwards-Wilkinson [11] and c = forthe single step (SS) model [38, 39]. For growth, we do nothave an equipartition theorem, but we have w ( t → ∞ ) = w s , thus we replace D by the above value to obtain (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = 2 bνw s δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) , (16)with b = 1 / ( cL ). Therefore, the parameter D in eq. (13) isnot only the noise intensity but it is also related to ν , in theabove equation, which leads to a fluctuation-dissipationtheorem (FDT). Note the similarity between eq. (16) andeq. (7). Also, since the noise and the surface tension in theEdwards-Wilkinson equation have their origin in the flux,the separation between them is artificial, consequently, theabove discussion restore the lost link.It is important to note that the only difference betweenthe Edwards-Wilkinson and the KPZ FDT is in the con-stant c . Considering that the growth phenomenon is muchmore complex than diffusion, this is an impressive result.Moreover, c depends only on the universality class, particu-larly for KPZ, it corresponds to a large number of differentmodels. Note as well that the dispersion of the velocityp-2itle N o r m a li z ed Q uan t i t y Normalized timeGrowth (etching model)Growth (EW model)Diffusion
Fig. 1: Dispersions as a function of time. For diffusion, wehave the mean squared velocity in units of (cid:112) k B T /m ; Forgrowth, the normalized roughness, w ( t ) /w s , as functionof their normalized time for the KPZ universality class(etching model) and for the Edwards-Wilkinson universalityclass (EW model).is proportional to T while the roughness is proportinal to L , which means that < v > eq ∝ T and w s ∝ L , respec-tively. Therefore, the size L is the “ingredient” that allowsdisorder in growth phenomena.Now, if we add a force in the Langevin equation, linearor not, it does not affect the FDT. The same for the EWequation: if we add, for example, λ (cid:126) ∇ h ( (cid:126)x, t )] , transform-ing it in the KPZ equation, it does not alter the FDT. Itis noteworthy that for the SS model the height differencebetween next neighbors is ±
1, then the [ (cid:126) ∇ h ( (cid:126)x, t )] is justa constant, which for a stronger reason it will not affectthe FDT, except for the constant b above. Consequently,there is an implicit FDT for the growth process, similarto the Langevin equation. This was the missing part thatneeded to be clarified. Besides, note that the equation (16)depends on Periodic Boundary Conditions (PBC) becauseboth Edwards-Wilkinson [11] and Krug et al. [38,39] resultinto eq. (15) was obtained using PBC.In this way, we show in Fig (1) the evolution of thedispersions with time. In the upper curve we exhibit themean square velocity ˜ v = (cid:112) (cid:104) v ( t ) (cid:105) in units of the “sat-urated” or mean squared equilibrium velocity (cid:112) k B T /m ,and time in units of normalized time t/τ with τ = γ − .The curve was obtained by solving eq. (1), with the ini-tial condition v ( t = 0) = 0, and averaging over 1 × numerical experiments.Simulations of the growth equations are usually morehard [16], thus we use cellular automaton model in thesame class of universality of the growth stochastic equation.In this way, the middle curve is for the Edwards-Wilkinsonmodel, we plot the roughness, w ( t ), normalized to theproper units, i.e w ( t ) /w s , as a function of normalized time t/t × . In order to model the EW equation we shall use theSS model. The SS model has gained a lot of importanceover the years [38–42]: firstly, because it was proved tobe a KPZ model and secondly due to its connection withother models, such as the asymmetric simple exclusionprocess [40], the six-vertex model [41,43,44], and the kineticIsing model [41, 45]. Finally, our interest here is becausethis model, in particular, can become an EW model. TheSS model is defined in such way that the height differencebetween two neighbors heights η = h i − h j is just η = ± i are of the form h i (0) = (1 + ( − i ) / L and we define the space Ω = 1 , , , · · · , L − , L . First, weselect a site i randomly, then we compare its height withits neighbors, and we apply the following rules:1. At time t , randomly choose a site i ∈ Ω;2. If h i ( t ) is a minimum, then h i ( t + ∆ t ) = h i ( t ) + 2,with probability p ;3. If h i ( t ) is a maximum, then h i ( t + ∆ t ) = h i ( t ) − q .With the above rules we can generate the dynamics ofthe SS model for d + 1 dimensions. For 1 + 1 dimensionsits properties have been well studied [38–40], and we canobtain analytically results such as λ = p − q. (17)Note that for p = q , it becomes the EW model. Chang-ing the probabilities we can get a lot of relevant infor-mation [42]. Now, we start to simulate the model with p = q = 1 /
2, i.e. λ = 0 (EW model), with L = 128 andthe initial condition h i (0) = (1 + ( − i ) /
2. We use theabove rules, periodic boundary conditions and we averageover 1 × numerical experiments to obtain the middlecurve with w s = 3 . t x = 514 . w ( t ), normalized toproper units, i.e w ( t ) /w s , as a function of normalized time t/t × , which was obtained from the 1 + 1 etching model.The etching model [13, 20, 21, 46] is a stochastic cellularautomaton that simulates the surface erosion due to actionof an acid. This model is proposed as simple as possible inorder to capture the essential physics, which considers thatthe probability of removing a cell is proportional to thenumber of the exposed faces of the cell (an approximationof the etching process). First, we randomly select a site i with a certain height, h i , and then we compare it withone of its nearest neighbor j . This is similar to the SSmodel, but the rules change: if h j > h i , it is reduced tothe same height as h i , which means that the height of thesurface decreases at each step. The main algorithm stepsare summarized as follows:1. At time t , randomly choose a site i ∈ V ;p-3omes-Filho et al.
2. If h j ( t ) > h i ( t ), then h j ( t + ∆ t ) = h i ( t );3. Consider h i ( t + ∆ t ) = h i ( t ) − . This stochastic cellular automaton has the advantage ofallowing us to understanding a corrosion process, and atsame time to study the KPZ growth process. Note that therule 1) is commom to all growth models, it is equivalentto the random term ξ ( (cid:126)x, t ) in the KPZ equation (12). Theinteraction between the neighbors is equivalent to diffusiveones, and the nonlinear term represents the lateral growth.The etching model belongs to the KPZ universality class [21,46]. Now we use these rules to simulate this model. Wetake the initial condition as h i ( t = 0) = 0, and we useperiodic boundary conditions. We take L = 128 and weaverage over 4 × numerical experiments, and we obtain w s = 7 . t x = 163 . Correlated noise.—
As exposed above, the FDT has animportant place in statistical mechanics. Not only becauseit is a basic statement, but also because it allows us to ob-tain important measured quantities, such as susceptibility,the light scattering cross section, the neutron scatteringintensity, diffusion, surface roughness in growth, and so on.We make now a generalization of the previous result forcorrelated noise.A natural extension of the Langevin formalism is theMori equation [47–53] for the operator ˆ O ( t ) as d ˆ O ( t ) dt = − (cid:90) t Γ( t − s ) ˆ O ( s ) ds + f ( t ) . (18)Now, the fluctuating force, f ( t ), obeys the following condi-tions:( i ) the mean force is zero, (cid:104) f ( t ) (cid:105) = 0 , (19)( ii ) there is no correlation between the initial value for theoperator and the random force, (cid:104) f ( t ) ˆ O (0) (cid:105) = 0 (20)and ( iii ) the fluctuating forces at different times t and t (cid:48) are correlated as (cid:104) f ( t ) f ( t (cid:48) ) (cid:105) = Γ( t − t (cid:48) ) . (21)The basic difference from the Langevin’s formalism isthat we now have a correlation and the above relation defines a memory function Γ( t ), and the Mori formalismis a Quantum formalism. Note that for the particularcase Γ( t ) = 2 γδ ( t ) we return to the Langevin’s formalism.Now we use the above conditions for the noise to obtain aself-consistent equation for the correlation function R ( t ), R ( t ) = (cid:104) ˆ O ( t ) ˆ O (0) (cid:105)(cid:104) ˆ O (0) (cid:105) , (22)namely dR ( t ) dt = − (cid:90) t R ( t − t (cid:48) )Γ( t (cid:48) ) dt (cid:48) . (23)The determination of correlation function is fundamentalfor the determination of the dynamics. Thus, the equa-tion (23) is an important part of the theory, which means,given Γ( t ) we can get R ( t ) and then all dynamics. For theMori equation, the correlation function has been subject ofintense investigation [50–58]. Correlated phenomena yieldvery peculiar forms of anomalous relaxation [6, 59–67].The equivalent of that for the growth phenomena is toconsider a correlated noise in the asymptotic form [10, 68]: (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) ∝ | x − x (cid:48) | φ − | t − t (cid:48) | θ − , (24)for the 1 + 1 KPZ equation. Here 0 < φ < . < θ < . φ and θ the growth exponents are the same as thoseof the uncorrelated, or strong local correlation [68, 69].However, for θ ≈ .
45, we have α ≈ . (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = 2 Dδ ( x − x (cid:48) ) | t − t (cid:48) | θ − , (25)which is more easy to connect with anomalous diffusion.For example, in this situation α (cid:54) = 1 / w s ∝ ( D/ν ) α ,and thus the eq. (25) becomes (cid:104) ξ ( x, t ) ξ ( x (cid:48) , t (cid:48) ) (cid:105) = C νw /αs δ ( x − x (cid:48) ) | t − t (cid:48) | θ − , (26)where C and α are not universal constants, i.e. theydepend on θ . Therefore, the eq. (26) is not universal asthe eq. (16). Violation of the FDT: the hierarchy.—
The violation ofthe FDT is a very common phenomenon. For example ithappens in ballistic diffusion [55–57], and in KPZ [12, 28]for d >
1. The violation of the FDT was also observed instructural glass [70–75], in proteins [76], and in mesoscopicradioactive heat transfer as well [77, 78]. Thus, an explicitFDT for the d + 1 KPZ equation, such as the above one for1 + 1 dimensions goes beyond our objective here. However,due to the importance of the FDT to understand why itfails is a must. The first step was to realize that there isa hierarchy in the major theorems of statistical physics.First, note that if lim t →∞ R ( t ) = 0 , (27)p-4itlenamely, the mixing condition (MC), holds, the systemforgets its initial conditions [61, 62] and then reaches a fullequilibrium. The Khinchin theorem [61,62] establishes thatif (27) holds then ergodicity holds as well. In anomalousdiffusion the violation of the FDT has been associated withthe violation of the mixing condition [55–57], which meanslim t →∞ R ( t ) = κ (28)with κ (cid:54) = 0, which was denominated as the non-ergodicfactor [55]. The sequence of works [55–57, 62] was veryimportant because it established in an exact and analyticalway the violation of the FDT and how it happens. Thehierarchy is simple, the Khinchin’s theorem is more im-portant than ergodicity, which is more important than theFDT. Unfortunately, we did not achieve such relation forthe d + 1 KPZ equation. Last remarks-—
The fluctuation is a very common phe-nomenon and, as such, it is universal. The “dissipation”,however, is more subtle. In diffusive motion is easily toaccept because it is the friction force that we are used to.In the growth phenomenon, it appears as a “friction” tothe roughness, which means that, since the noise createslocal irregularities, the smoothing mechanism acts with a“frictional force” proportional to the local curvature, butwith an opposite sign to it, which reduces the roughness.The process continues until a saturated roughness w s isreached. Conclusion.—
In this work, we start discussing the el-egant formalism of Langevin to describe the Brownianmotion, where fluctuation-dissipation theorem (FDT) wasexplicitly stated for the first time [5]. The FDT establishesa clear connection between noise intensity (strength ofthe random force) and dissipation for a system in thermalequilibrium. Next, we present some types of Langevinequations, stochastic partial differential equations, suchas the Edwards-Wilkinson and the Kardar-Parisi-Zhangequation in 1 + 1 dimensions to describe the growth dy-namics. From this, we were able to show that there is ahidden FDT for the growth phenomenon, similar to thediffusive one, in the sense that the parameter D (noiseintensity) that increases the roughness is directly relatedto ν (surface tension),which decreases it. This gives us ex-plicitly a fluctuation-dissipation theorem. Furthermore, wenote that different classes of universality exhibit the samebehavior, with distinction only in the non-dimensional con-stant. We also extended our discussion to systems withcorrelated noise. We expect that this work will stimulatenew investigations in both correlated and higher dimensionsystems. ∗ ∗ ∗ This work was supported by the Conselho Nacional deDesenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), GrantNo. CNPq-312497/2018-0 and the Funda¸c˜ao de Apoio aPesquisa do Distrito Federal (FAPDF), Grant No. FAPDF-00193-00000120/2019-79. (F.A.O.).
REFERENCES[1]
Brown R. , Phil. Mag. , (1828) 161.[2] Brown R. , Ann. Phys. Chem. B , (1828) 294.[3] Einstein A. , Ann. Phys. , (1905) 549.[4] Einstein A. , Investigations on the theory of the BrownianMovement (Dover, New York) 1956.[5]
Langevin P. , C. R. Acad. Sci. (Paris) , (1908) 530.[6] Vainstein M. H., Costa I. V. L. and
Oliveira F. A. , Mixing, ergodicity and the fluctuation-dissipation theoremin complex systems in Jamming, Yielding, and IrreversibleDeformation in Condensed Matter , edited by
MiguelM. C. and
Rub´ı M. , Lect. Notes Phys. , Vol. 688 (Springer,Berlin, Heidelberg) 2006, 159–188.[7]
Gudowska-Nowak E., Lindenberg K. and
MetzlerR. , J. Phys. A , (2017) 380301. http://stacks.iop.org/1751-8121/50/i=38/a=380301 [8] Oliveira F. A., Ferreira R. M. S., Lapas L. C. and
Vainstein M. H. , Frontiers in Physics , (2019) 18.[9] Hansen A., Schmittbuhl J., Batrouni G. G. and de Oliveira F. A. , Geophys. Res. Lett. , (2000) 3639.[10] Barab´asi A. L. and
Stanley H. E. , Fractal Concepts inSurface Growth (Cambridge University Press, Cambridge)1995.[11]
Edwards S. F. and
Wilkinson D. R. , Proc. R. Soc.Lond. A , (1982) 17.[12] Kardar M., Parisi G. and
Zhang Y.-C. , Phys. Rev.Lett. , (1986) 889. http://link.aps.org/doi/10.1103/PhysRevLett.56.889 [13] Mello B. A., Chaves A. S. and
Oliveira F. A. , Phys.Rev. E , (2001) 041113. http://link.aps.org/doi/10.1103/PhysRevE.63.041113 [14] ´Odor G., Liedke B. and Heinig K.-H. , Phys. Rev. E , (2010) 031112.[15] Merikoski J., Maunuksela J., Myllys M., TimonenJ. and
Alava M. J. , Phys. Rev. Lett. , (2003) 24501.[16] Takeuchi K. A. , Phys. Rev. Lett. , (2013) 210604.[17] Almeida R. A. L., Ferreira S. O., Ferraz I. and
Oliveira T. J. , Sci. Rep. , (2017) 3773.[18] Aar˜ao Reis F. D. A. , Phys. Rev. E , (2005) 032601. http://link.aps.org/doi/10.1103/PhysRevE.72.032601 [19] Almeida R. A. L., Ferreira S. O., Oliveira T. J. and
Aar˜ao Reis F. D. A. , Phys. Rev. B , (2014)045309.[20] Rodrigues E. A., Mello B. A. and
Oliveira F. A. , J. Phys. A , (2015) 35001.[21] Alves W. S., Rodrigues E. A., Fernandes H. A.,Mello B. A., Oliveira F. A. and
Costa I. V. L. , Phys. Rev. E , (2016) 042119.[22] Carrasco I. S. S. and
Oliveira T. J. , Phys. Rev. E , (2018) 010102.[23] Dasgupta C., Sarma S. D. and
Kim J. , Physical ReviewE , (1996) R4552.[24] Dasgupta C., Kim J., Dutta M. and
Sarma S. D. , Physical Review E , (1997) 2235.[25] Torres M. and
Buceta R. , Journal of Statistical Me-chanics: Theory and Experiment , (2018) 033208.[26] Wio H. S., Revelli J. A., Deza R. R., Escudero C. p-5omes-Filho et al. and de la Lama M. S. , EPL (Europhysics Letters) , (2010) 40008. http://stacks.iop.org/0295-5075/89/i=4/a=40008 [27] Wio H. S., Rodr´ıguez M. A., Gallego R., RevelliJ. A., Al´es A. and
Deza R. R. , Frontiers in Physics , (2017) 52.[28] Rodr´ıguez M. A. and
Wio H. S. , Physical Review E , (2019) 032111.[29] Bertini L. and
Giacomin G. , Commun. Math. Phys. , (1997) 571.[30] Baik J., Deift P. and
Johansson K. , Journal of theAmerican Mathematical Society , (1999) 1119.[31] Pr¨ahofer M. and
Spohn H. , Phys. Rev. Lett. , (2000) 4882. https://link.aps.org/doi/10.1103/PhysRevLett.84.4882 [32] Dotsenko V. , Journal of Statistical Mechanics: Theoryand Experiment , (2010) P07010.[33] Calabrese P., Le Doussal P. and
Rosso A. , EPL(Europhysics Letters) , (2010) 20002.[34] Amir G., Corwin I. and
Quastel J. , Communicationson pure and applied mathematics , (2011) 466.[35] Sasamoto T. and
Spohn H. , Phys. Rev. Lett. , (2010) 230602. http://dx.doi.org/10.1103/PhysRevLett.104.230602 [36] Le Doussal P., Majumdar S. N., Rosso A. and
Schehr G. , Phys. Rev. Lett. , (2016) 070403.[37] Hairer M. , Ann. Math. , (2013) 559. http://annals.math.princeton.edu/2013/178-2/p04 [38] Krug J. , Adv. Phys. , (1997) 139-282.[39] Krug J., Meakin P. and
Halpin-Healy T. , PhysicalReview A , (1992) 638-653.[40] Derrida B. and
Lebowitz J. L. , Physical review letters , (1998) 209.[41] Meakin P., Ramanlal P., Sander L. M. and
BallR. , Physical Review A , (1986) 5091.[42] Daryaei E. , Physical Review E , (2020) 062108.[43] Gwa L.-H. and
Spohn H. , Physical review letters , (1992) 725.[44] De Vega H. and
Woynarovich F. , Nuclear Physics B , (1985) 439.[45] Plischke M., R´acz Z. and
Liu D. , Physical Review B , (1987) 3485.[46] Gomes W. P., Penna A. L. A. and
Oliveira F. A. , Phys. Rev. E , (2019) 020101. https://link.aps.org/doi/10.1103/PhysRevE.100.020101 [47] Mori H. , Prog. Theor. Phys. , (1965) 423.[48] Mori H. , Prog. Theor. Phys. , (1965) 399.[49] Kubo R., Toda M. and
Hashitsume N. , StatisticalPhysics II (Springer, Berlin) 1991.[50]
Lee M. H. , Phys. Rev. Lett. , (1983) 1227.[51] Lee M. H. , Phys. Rev. B , (1982) 2547.[52] Lee M. H. and
Hong J. , Phys. Rev. B , (1984) 6756.[53] Florencio J. and
Lee M. H. , Phys. Rev. A , (1985)3231.[54] Morgado R., Oliveira F. A., Batrouni G. G. and
Hansen A. , Phys. Rev. Lett. , (2002) 100601.[55] Costa I. V. L., Morgado R., Lima M. V. B. T. and
Oliveira F. A. , Europhys. Lett. , (2003) 173.[56] Lapas L. C., Costa I. V. L., Vainstein M. H. and
Oliveira F. A. , Europhys. Lett. , (2007) 37004.[57] Lapas L. C., Morgado R., Vainstein M. H., Rub´ıJ. M. and
Oliveira F. A. , Phys. Rev. Lett. , (2008)230602.[58] Florˆencio J. and de Alcantara Bonfim O. F. , Fron-tiers in Physics , (2020) 507. doi:10.3389/fphy.2020.557277 [59] Kohlrausch R. , Ann. Phys , (1854) 179.[60] Kohlrausch F. , Ann. Phys , (1863) 337.[61] Khinchin A. I. , Mathematical Foundations of StatisticalMechanics (Dover, New York) 1949.[62]
Lee M. H. , Phys. Rev. Lett. , (2007) 190601.[63] Vainstein M. H., Costa I. V. L., Morgado R. and
Oliveira F. A. , Europhys. Lett. , (2006) 726.[64] Lapas L. C., Ferreira R. M., Rub´ı J. M. and
Oliveira F. A. , J. Chem. Phys. , (2015) 104106.[65] Weron A. and
Magdziarz M. , Phys. Rev. Lett. , (2010) 260603. https://link.aps.org/doi/10.1103/PhysRevLett.105.260603 [66] Dybiec B., Parrondo J. M. R. and
Gudowska-Nowak E. , EPL , (2012) 50006. http://stacks.iop.org/0295-5075/98/i=5/a=50006 [67] Costa, I. V. L., Vainstein M. H., Lapas L. C.,Batista. A. A. and
Oliveira F. A. , Phys. A: Stat.Mech. Appl. , (2006) 130. https://doi.org/10.1016/j.physa.2006.04.096 [68] Lam Chi-Hang, Sander L. M., and
Wolf D. E. , Phys.Rev. A , (1992) R6128.[69] Medina E., Hwa T., Kardar M. and
Zhang Yi-Cheng , Phys. Rev. A , (1989) 3053.[70] Grigera T. S. and
Israeloff N. E. , Phys. Rev. Lett. , (1999) 5038.[71] Ricci-Tersenghi F., Stariolo D. A. and
ArenzonJ. J. , Phys. Rev. Lett. , (2000) 4473.[72] Crisanti A. and
Ritort F. , Journal of Physics A:Mathematical and General , (2003) R181.[73] Barrat A. , Physical Review E , (1998) 3629.[74] Bellon L. and
Ciliberto S. , Physica D: NonlinearPhenomena , (2002) 325.[75] Bellon L., Buisson L., Ciccotti M., Ciliberto S. and
Douarche F. , Thermal noise properties of two ag-ing materials arXiv:cond-mat/0501324 [cond-mat.stat-mech](2005).[76]
Hayashi K. and
Takano M. , Biophysical journal , (2007) 895.[77] P´erez-Madrid A., Lapas L. C. and
Rub´ı J. M. , Phys.Rev. Lett. , (2009) 048301. https://link.aps.org/doi/10.1103/PhysRevLett.103.048301 [78] Averin D. V. and
Pekola J. P. , Phys. Rev. Lett. , (2010) 220601. https://link.aps.org/doi/10.1103/PhysRevLett.104.220601https://link.aps.org/doi/10.1103/PhysRevLett.104.220601