Anomalous Heat Equation in a System Connected to Thermal Reservoirs

Anomalous heat equation in a system connected to thermal reservoirs

Priyanka,∗ Aritra Kundu,† Abhishek Dhar,‡ and Anupam Kundu§ International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India (Dated: March 20, 2018) We study anomalous transport in a one-dimensional system with two conserved quantities in presence of thermal baths. In this system we derive exact expressions of the temperature profile and the two point correlations in steady state as well as in the non-stationary state where the later describe the relaxation to the steady state. In contrast to the Fourier heat equation in the diffusive case, here we show that the evolution of the temperature profile is governed by a non-local anomalous heat equation. We provide numerical verifications of our results.

Transport of energy across an extended system is a — we refer to this model as the harmonic chain momen- paradigm of the vast class of non-equilibrium phenom- tum exchange (HCME) model. For the infinite HCME ena. At a macroscopic level this phenomena is often system it was shown exactly that the current autocorre- 1/2 described by the phenomenological Fourier’s law which lation has a t− decay [17]. It was also shown that, relates the energy current density j(x, t) to the gradient in contrast to∼ Eq. (1), the evolution of an initially local- of the temperature field T (x, t): j = κ ∂xT where κ ized energy perturbation satisfied a non-local fractional − 3/4 is the thermal conductivity. This law implies diffusive diffusion equation ∂te(x, t) = c( ∆) e(x, t), where energy flow across the system described by the Fourier e(x, t) is the energy perturbation− and− c is some constant heat equation [12–14, 18]. The fractional laplacian operator ( ∆)3/4 in the infinite space is defined by its Fourier spectrum:− 2 3/2 ∂τ T (x, t) = D ∂xT (x, t), (1) q whereas the same for the normal Laplacian opera- | | 2 2 3/4 tor ∆ ∂x is q . In real space ( ∆) operator is where D = κ/c (for simplicity we assume κ and the spe- non-local− ≡ [19, − 20]. − cific heat c to be independent of temperature). This While all these studies consider transport in isolated equation is widely used in experiments to understand systems, quite often the transport set up in an exper- the spreading of local energy perturbations in equilib- iment consists of an extended system connected at the rium as well as the non-equilibrium dynamics of systems two ends to heat baths at different temperatures. For connected to reservoirs. diffusive systems Eq. (1) continues to describe both non- Surprisingly, several theoretical [1–3], numerical as well equilibrium steady state (NESS) and time-dependent as experimental studies [4] suggest that in many one and properties in this set-up. It is then natural to ask: what two dimensional systems heat transfer is anomalous in would be the corresponding evolution equation for the the sense that Fourier’s law is not valid [5–7]. This temperature profile in the case of anomalous transport phenomenon is usually manifested by several interesting in the experimental set-up? A major problem that now features like divergence of thermal conductivity κ with arises follows from the fact that the fractional Laplacian system size L as κ Lα; 0 < α < 1, power-law de- is a non-local operator and so extending its definition to cay of the equilibrium∼ current-current auto-correlations, a finite domain is non-trivial. Several studies have ad- super-diffusive spreading of local energy perturbations, dressed this issue, using a phenomenological approach, nonlinear stationary temperature profiles (even for small in the context of Levy walks and Levy flights in finite temperature differences) and the presence of boundary domains [21, 22]. It is thus crucial to have examples of singularities in these profiles [1, 6–14]. specific microscopic models of systems exhibiting anoma- There is currently no general framework to describe lous transport, for which the time-evolution equation in and explain anomalous heat transport. Recently, the the- an open system set up can be derived analytically, and ory of nonlinear fluctuating hydrodynamics has been re- where one can see the non-local and fractional equation markably successful in predicting anomalous scaling of forms explicitly. This is the main aim of this Letter. dynamical correlations of conserved quantities in one- Such attempts have recently been made in [23–25] arXiv:1803.06857v1 [cond-mat.stat-mech] 19 Mar 2018 dimensional Hamiltonian systems and the corresponding where the problem of non-linear steady state tempera- slow decay of the equilibrium current-current autocorre- ture profiles and their time-evolution in the HCME model lations [2, 3, 15, 16]. This approach provides diverging was addressed. The specific model studied was a har- thermal conductivity (via Green-Kubo formula) as well monic chain of N particles where, in addition to the as Lévy scaling for the spreading of local energy perturba- Hamiltonian dynamics, the momenta of nearest neigh- tion. On the rigorous side, computations were done for borhood particles is exchanged randomly at a constant a model of harmonic chain whose Hamiltonian dynam- rate γ. The chain is attached to two Langevin baths at ics was supplemented by a stochastic part that kept the the two ends at temperatures T` and Tr. This system has conservation laws (number, energy, momentum) intact three conserved fields: the stretch r = q q (where i i+1 − i 2 qi, i = 1,...,N are the particle positions) the momen- and η2j = pj, one finds that both the above equations 2 2 tum, pi and the energy i = pi /2 + ri /2. This system can be expressed in a single equation: η˙m = ηm+1 ηm 1 1/2 − − shows anomalous current behavior j N − as well for m = 1, 2, ..., L. The system can also be interpreted as exhibits a non-linear stationary temperature∼ profile as a fluctuating interface where the algebraic volume of 2 Ti = pi /2 ss = (i/N), which was computed analyti- the interface at site m is given by ηm and the energy cally forh fixedi andT free boundary conditions — surpris- V (η) = η2/2 [13]. Hence, the stochastic exchange part in ingly the temperature-profile was different for these cases Eq. (2) can be thought of as a volume-energy conserving [23, 25]. The evolution of the non-stationary tempera- noise. We call this model as ‘harmonic chain with volume ture profile (x, τ) (where the τ = t/N 3/2 is the rescaled exchange’ (HCVE). time) to theT NESS profile was also studied [26], where by It has been shown that the HCVE model defined on eliminating the fast variables it was shown that (x, τ) an isolated infinite one dimensional lattice (i.e. λ = 0 T satisfies an energy continuity equation. From an analysis in Eq. (2) with i = , .., 1, 0, 1, .., ) exhibits super of this equation it was noted that the evolution appears diffusion of energy [18]:−∞ − ∞ to be similar to the fractional diffusion equation. How- ever, so far this has not been clearly established and in ∂te(x, t) = L [e(x, t)], − ∞ particular an explicit representation of the correspond- 1 3/4 1/4 (3) L = [( ∆) ( ∆) ], ing fractional evolution operator is not known. In this ∞ √2γ − − ∇ − Letter, we look at a simpler model of anomalous transport in one dimension where we derive the corresponding where the skew-fractional operator L has the Fourier ∞ fractional evolution equation for the temperature profile representation q 3/2(1 i sgn(q)) with i = √ 1 and | | − − inside a finite domain and show explicitly how this evo- sgn(q) is the Signum function. In this paper, however, lution approaches to the appropriate fractional diffusion we consider the HCVE model on a finite lattice of size L operator in the infinite domain. in open set up i.e. connected to heat baths at the two This model consists of a finite one dimensional lat- ends as described in Eq. (2). It is known that in this tice of L sites where each site carries a ‘stretch’ vari- case also, as in HCME, the stationary current scales as 1/2 able ηi, i = 1, 2, ..., L under an onsite external potential j L− [18]. 2 ∼ V (ηi) = ηi /2. The lattice is attached to two thermal Results.- We explicitly find that in the large L limit reservoirs at temperatures T` and Tr on the left and right the stationary ‘energy’ current jss is given, in the leading ends, respectively and subjected to a volume conserving order, by stochastic noise. The dynamics of this model has two r parts: (a) the usual deterministic part plus the Langevin 1 π (T` Tr) jss = − . (4) terms coming from the baths and (b) a stochastic ex- 2 γ √L change part where ηs from any two neighboring sites, In the non-stationary regime, we numerically find that chosen at random, are exchanged at some rate γ. The the temperature profile and the two- dynamics is given by Ti(t) = V (ηi(t)) point correlations C (t) = η (t)hη (t) forii = j have the i,j h i j i 6 dηi following scaling forms = V 0(ηi+1) V 0(ηi 1) dt − − p i t + δ λV 0(η ) + 2λT ζ (t) Ti(t) = , i,1 − 1 ` ` (2) T L L3/2 p (5) + δi,L λV 0(ηL) + 2λTrζr(t) 1 i j i + j t − Ci,j(t) = | − |, , , √LC √L 2L L3/2 + stochastic exchange at rate γ in the leading order for large L. The scaling functions with fixed boundary conditions (BCs) η0 = ηL+1 = 0. (y, τ) and (x, y, τ) satisfy the following equations in- Here ζ`,r(t) are mean zero and unit variance, indepen- Tside the domainC = 0 x ; 0 y 1 : dent Gaussian white noises. Note that, in contrast to D { ≤ ≤ ∞ ≤ ≤ } the HCME case, this dynamics has two conserved quan- 2 ∂y (x, y, τ) = γ∂x (x, y, τ) (6) tities: the ‘volume’ η and the energy V (η ). This model C − C i i ∂ (y, τ) = 2γ [∂ (x, y, τ)] (7) was first introduced by Bernardin and Stoltz in the closed yT − xC x=0 ∂ (y, τ) = 2∂ (0, y, τ), (8) system setup [13] where starting from the harmonic chain τ T yC with Hamiltonian given earlier, they have treated the with (x, y, 0) x = 0 and (x, y, 0) = 0. We find that positions qis and the momenta pis on the same foot- C | →∞ C ing. Note that for harmonic chain, the dynamics of the the exact solutions of these equations are given by ‘stretch’ variable ri = qi+1 qi and the momentum vari- − (y, τ) = ss(y) + r(1 y, τ) (9) able are similar: r˙i = pi+1 pi and p˙i = ri+1 ri for T T T − − − (x, y, τ) = ss(x, y) + r(x, 1 y, τ). (10) i = 1, 2, ..., N. Hence for N = L/2, defining η2j 1 = rj − C C C − 3

In the above equation, NESS part of the proﬁles are 1.10 =1.0 p =2.0 ss(y) = Tr + (T` Tr) 1 y, 1.05 T − − Theory r ! (11) T` Tr π x ) y

ss(x, y) = − erfc , y p ( 1.00 0.0 0.2 0.4 0.6 0.8 1.0 C − 4 γ 4γ(1 y) 0.00 ss x =1.0, =1.0

− AAAB/XicbVDNSsNAGNzUv1r/ouLJy2Ir1EtJelEvUvDisUJjC20Im+2mXbrZhN2NEELAV/HiQcWr7+HNt3HT5qCtAwvDzPfxzY4fMyqVZX0blbX1jc2t6nZtZ3dv/8A8PHqQUSIwcXDEIjHwkSSMcuIoqhgZxIKg0Gek789uC7//SISkEe+pNCZuiCacBhQjpSXPPGmMQqSmGLGsl3uZlHkzvWh4Zt1qWXPAVWKXpA5KdD3zazSOcBISrjBDUg5tK1ZuhoSimJG8NkokiRGeoQkZaspRSKSbzePn8FwrYxhEQj+u4Fz9vZGhUMo09PVkkVUue4X4nzdMVHDlZpTHiSIcLw4FCYMqgkUXcEwFwYqlmiAsqM4K8RQJhJVurKZLsJe/vEqcduu6Zd23652bso0qOAVnoAlscAk64A50gQMwyMAzeAVvxpPxYrwbH4vRilHuHIM/MD5/ABqulRs= T ) -0.01 N = 1024 and the relaxation parts are x, y 0.02 0.95 ( - N = 2048 ss AAAB/3icbVBNS8NAEN3Ur1q/oh48eFlshQpSkl60t0IvHisYW2hD2Gy37dLNJuxuxBBy8a948aDi1b/hzX/jps1BWx8MPN6bYWaeHzEqlWV9G6W19Y3NrfJ2ZWd3b//APDy6l2EsMHFwyELR95EkjHLiKKoY6UeCoMBnpOfPOrnfeyBC0pDfqSQiboAmnI4pRkpLnnlSGwZITTFiaSfzUimz+uNlclHzzKrVsOaAq8QuSBUU6Hrm13AU4jggXGGGpBzYVqTcFAlFMSNZZRhLEiE8QxMy0JSjgEg3nT+QwXOtjOA4FLq4gnP190SKAimTwNed+bVy2cvF/7xBrMbXbkp5FCvC8WLROGZQhTBPA46oIFixRBOEBdW3QjxFAmGlM6voEOzll1eJ02y0GtZts9puFWmUwSk4A3VggyvQBjegCxyAQQaewSt4M56MF+Pd+Fi0loxi5hj8gfH5A1F5lb0= C -0.03 x2 z -0.04 Z exp 4γ(z z0) ∂ (z , τ) − − r 0 0.90 r(x, z, τ) = T dz0 (12) p 0.0 0.2 0.4 0.6 0.8 1.0 C − 0 4πγ(z z0) ∂z0 y − where r(z, τ) satisfies the following continuity equation: FIG. 1. Numerical verification of the analytical NESS pre- T dictions for Tss(y) and Css(x, y) in Eq. (11). Symbols are ob- Z z 1 ∂z0 r(z0, τ) tained from simulations with ω = γ = 1,T` = 1.1,Tr = 0.9 ∂τ r(z, τ) = ∂z dz0 T , (13) and N = 1024, whereas the solid lines are from theory. T √πγ 0 √z z0 − =1.0, ! = =1,N= 1024 inside the domain 0 z 1 with BCs r(0, τ) = =2.0, ! = =1,N= 1024 ≤ ≤ T is usually hard to compute explicitly the full joint distri- r(1, τ) = 0. The relaxation parts r(z, τ) and r(x, z, τ) describeT the approach towards theT NESS solutionsC in the bution P (~η, t), computing various correlations and fluc- τ limit. The equations Eq. (4), Eq. (11), Eq. (12) tuations is more tractable and they contain the most use- and→ Eq. ∞ (13), comprise our main results. Note that the ful information about the non-equilibrium state. In this evolution of the temperature in Eq. (13) is indeed given paper, for HCVE model we compute all the two point correlations η (t)η (t) exactly in the limit of large sys- by a linear but non-local equation defined inside a finite h i j i domain 0 z 1. However, following a similar cal- tem size. culation for≤ infinite≤ system we later show that Eq. (13) Starting from the FP equation in Eq. (14), we obtain 2 reduces to Eq. (3) [27]. This establishes, without am- the dynamical equations satisfied by Ti = ηi (t) and C = η (t)η (t) for i = j in the bulk: h i biguity, that the non-local operator in Eq. (13) is the i,j h i j i 6 correct finite domain representation of the fractional op- C˙ ij = Ci+1,j Ci 1,j + Ci,j+1 Ci,j 1 erator L in Eq. (3). Another point to note that the − − ∞ − − temperature profile in SS, (y), is asymmetric under + γ[Ci 1,j + Ci+1,j + Ci,j 1 + Ci,j+1 4Ci,j], ss − − − space reversal as the microscopicT model itself does not C˙ i,i+1 = Ti+1 Ci 1,i+1 + Ci,i+2 Ti (15) have such symmetry. As a result, any locally created − − − + γ[Ci 1,i+1 + Ci,i+2 2Ci,i+1], perturbation splits into one traveling sound mode and − − one non-moving heat mode. This is in contrast to the T˙i = 2[Ci,i+1 Ci 1,i] + γ[Ti+1 + Ti 1 2Ti]. − − − − HCME model where one observes two sound modes moving in opposite directions in addition to a non-moving Rest of the equations at the boundaries are given in [27]. heat mode [13, 15]. Consequently, in this case, there is Fortunately the equations for two point correlations do not involve higher order correlations, this allows us to singularity in ∂y ss(y) only at one boundary and we find T solve these equations analytically, in the L limit. that the meniscus exponent [28] is again 1/2 as in the → ∞ HCME model with fixed boundary conditions. Interest- To proceed we follow the strategy in [26]. We first ingly, it turns out that for this boundary condition, both solve these equations numerically to observe that, for the temperature and the correlation become independent large L the solutions have the scaling properties as given of the strength of coupling λ with the heat baths in the in Eq. (5) where we have two length scales of O(L) along large L limit. the diagonal (i + j =constant) and of O(√L) along per- Derivation of the results: We start with the Fokker- pendicular to the diagonal ( i j =constant) direction, 3/2| − | Planck (FP) equation associated to the dynamics Eq. (2), and a time scale of O(L ). This time scale can be q 3/2[1 isgn(q)]t which describes the evolution of the joint distribution anticipated from the propagator e−| | − of the P (~η, t) of ~η = (η , η , , η ) at time t : Eq. (3) in Fourier space. The two length scales are un- 1 2 ··· L derstood by looking at the orders of the Ci,j and Ti, ∂tP (~η, t) = [ ` + b + ex] P (~η, t), (14) and their derivatives numerically [27]. These observa- L L L tions suggest that we look for solutions of Eq. (15) in the where, ` is the Liouvillian part, b contains the effects scaling form Eq. (5). Inserting these forms in Eq. (15), L L 1/2 of the Langevin baths at the boundaries and ex repre- and expanding in = L− , we obtain three coupled lin- sents the contribution from the exchange noise.L Explicit ear differential equations as given in Eq. (6)-Eq. (8) in the expressions of these operators are given in [27]. While it leading order [27]. Interestingly, the scaled correlation 4

1.10 satisﬁes the following equations τ=0.1

τ=0.2 2 ∂z r(x, z, τ) = γ∂x r(x, z, τ), (16) 1.05 τ=0.3 C C ∂z r(z, τ) = 2γ [∂x r(x, z, τ)]x=0 , (17) 0.00 SS T C )

⌧ ∂τ r(z, τ) = 2∂z r(0, z, τ), (18) 1.00 -0.02 T − C y, ) ( ⌧ x =0.25 AAAB/XicbVDLSsNAFJ34rPUVFVduBluhgpSkG+2u4MZlhcYWmlBuppN26OTBzEQooeCvuHGh4tb/cOffOGmz0NYDA4dz7uWeOX7CmVSW9W2srW9sbm2Xdsq7e/sHh+bR8YOMU0GoQ2Iei54PknIWUUcxxWkvERRCn9OuP7nN/e4jFZLFUUdNE+qFMIpYwAgoLQ3M06obghoT4FlnVpteuQrSy+rArFh1aw68SuyCVFCB9sD8cocxSUMaKcJByr5tJcrLQChGOJ2V3VTSBMgERrSvaQQhlV42jz/DF1oZ4iAW+kUKz9XfGxmEUk5DX0/mWeWyl4v/ef1UBTdexqIkVTQii0NByrGKcd4FHjJBieJTTYAIprNiMgYBROnGyroEe/nLq8Rp1Jt1675RaTWLNkroDJ2jGrLRNWqhO9RGDiIoQ8/oFb0ZT8aL8W58LEbXjGLnBP2B8fkDc/mUqw== T -0.04 with initial condition r(x, z, 0) = 0 and BC

x, y, C 0.95 ( -0.06 r(x, z, τ) x = 0. The above equations are obtained AAAB/3icbVBNS8NAEN34WetX1IMHL4utUKGUpBftrdCLxwrGFppQNtttu3SzCbsbMYRc/CtePKh49W9489+4aXPQ1gcDj/dmmJnnR4xKZVnfxtr6xubWdmmnvLu3f3BoHh3fyzAWmDg4ZKHo+0gSRjlxFFWM9CNBUOAz0vNnndzvPRAhacjvVBIRL0ATTscUI6WloXladQOkphixtJPVHutJ3VUovqwOzYrVsOaAq8QuSAUU6A7NL3cU4jggXGGGpBzYVqS8FAlFMSNZ2Y0liRCeoQkZaMpRQKSXzh/I4IVWRnAcCl1cwbn6eyJFgZRJ4OvO/Fq57OXif94gVuNrL6U8ihXheLFoHDOoQpinAUdUEKxYognCgupbIZ4igbDSmZV1CPbyy6vEaTZaDeu2WWm3ijRK4AycgxqwwRVogxvQBQ7AIAPP4BW8GU/Gi/FufCxa14xi5gT8gfH5A6r/lVI= C C | →∞ -0.08 from Eq. (6)-(8) after subtracting the steady state part 0.0 0.2 0.4 0.6 0.8 1.0 and then making the variable transformation z = (1 y).

1AAAB63icbVBNS8NAEJ3Ur1q/qh69LLaCF0tSBPVW8OKxgrGFNpTNdtMu3d2E3Y0QQn+DFw8qXv1D3vw3btsctPXBwOO9GWbmhQln2rjut1NaW9/Y3CpvV3Z29/YPqodHjzpOFaE+iXmsuiHWlDNJfcMMp91EUSxCTjvh5Hbmd56o0iyWDyZLaCDwSLKIEWys5Ne9i6w+qNbchjsHWiVeQWpQoD2ofvWHMUkFlYZwrHXPcxMT5FgZRjidVvqppgkmEzyiPUslFlQH+fzYKTqzyhBFsbIlDZqrvydyLLTORGg7BTZjvezNxP+8Xmqi6yBnMkkNlWSxKEo5MjGafY6GTFFieGYJJorZWxEZY4WJsflUbAje8surxG82bhrufbPWuizSKMMJnMI5eHAFLbiDNvhAgMEzvMKbI50X5935WLSWnGLmGP7A+fwB4YmNjg== y 0.90 Note that the BC in Eq. (17) acts like a current source,− 0.0 0.2 0.4 0.6 0.8 1.0 y at x = 0 boundary, to the diffusion Eq. (16). It is easy to show [27] that the solution of this equation is given FIG. 2. Numerical verification of the evolution of the temper- precisely by Eq. (12). Now, inserting this solution in ature profiles T (y, τ) = Tss(y)+Tr(1−y, τ) obtained using the solution of Eq. (13). Inset shows the verification of the corre- Eq. (18) and performing some simplifications we arrive lation C(x, y, τ) = Css(x, y) + Cr(x, 1 − y, τ) given in Eq. (10) at the non-local evolution Eq. (13) (see [27]) with bound- where C is computed using the solution in Eq. (12). The ary conditions (0, τ) = (1, τ) = 0. Hence it is natural r Tr Tr green line and the dashed purple line represent the initial and to expand the solution in sin basis αn(z) = √2 sin(nπz), the NESS temperature profiles, respectively. Symbols are ob- P ˆ n = 1, 2, ...: r(z, τ) = n∞=1 r(n, τ) αn(z) for all τ 0. tained from simulations with λ = γ = 1,T` = 1.1,Tr = 0.9 Inserting thisT form in Eq. (13)T and simplifying we≥ find and L = 2048, and the solid lines are from theory. that the coefficients ˆr(n, τ) satisfy the following matrix equation T

d ˆr function relaxes very fast over much shorter time scale T = κB ˆr, (19) [O(L)] comparedC to the evolution time scale [O(L3/2)] of dτ T the temperature field . Due to this fact, Eq. (6) and where B is an infinite order matrix with elements Bm,n T Eq. (7) do not involve the time derivative. As a result specified in [27] and κ = 1/√πγ. While it is difficult to the correlation function evolves adiabatically obeying solve this infinite order matrix equation analytically, we the (anti-)diffusion Eq. (6),C with a drive at the boundary solve it numerically by truncating it at some finite order. by the time dependent temperature field through Eq. (7). In Fig. 2, we compare the evolution from this numerical The equation for the temperature profile given in Eq. (8) solution with the same obtained from direct numerical is in the expected continuity equation. simulation of Eq. (2) and observe nice agreement. Using this solution in Eq. (12) we obtain C(x, z, τ) in Eq. (10) In the NESS the equations (6)-(8) become simpler since which we also compare with simulation results in the in- ∂ 0 as τ implying (0, y) = d. Now making τ ss set of Fig. 2 and again observe good agreement. theT variable → transformation→ ∞ zC= (1 y), the problem of From the numerical solution of Eq. (19) we, in addi- finding reduces to solving a diffusion− equation with its ss tion, observe that the eigenvalues obtained , are in gen- value atCx = 0 held fixed for all y. It is easy to show that p eral complex and are of the form λ nπ 3/2(1 isgn), the solution is given by (x, y) = d erfc(x/ 4γ(1 y)) n ss for large n > 0, while at small values≈ of | n| the eigenval-− where erfc(v) is the complementaryC error function− [27]. ues deviate from this behaviour [27]. We confirm that Now inserting this solution in Eq. (7) and solving with this is not due to the the truncation of the matrix but boundary conditions (0) = T and (1) = T , we ss ` ss r an artefact of the finiteness of the system. Note that the get the explicit expressionT Eq. (11). InT Fig. 1 we verify large n behavior of λ is similar to the Fourier spectrum the analytical results for T and numerically, where n ss ss of the non-local operator in Eq. (3) describing the we observe nice agreement. OneC can easily identify the L evolution in infinite system.∞ Hence it is interesting to see microscopic current from the equation for T˙ in Eq. (15) i if one recovers the evolution Eq. (3) in the infinite sys- as j = 2C γ(T T ) which in the steady state − i,i+1 − i+1 − i tem limit. One can follow the same calculation procedure for large provides √ . Note that the L jss = 2 ss(0, y)/ L on an infinite lattice as presented before for finite lattice term contributes− C at . Now inserting γ(Ti+1 Ti) (1/L) and arrive at an evolution equation for temperature pro- the expression− − of from Eq.O (11), one obtains the ss(0, y) file similar to Eq. (13) with only difference being that the expression for Cgiven in Eq. (4). jss lower limit of the integral on the rhs is , as the equa- We now focus on the relaxation to the NESS. It is tion is now valid in y . Now−∞ taking Fourier often convenient to separate the relaxation part as done transform on both sides−∞ it ≤ is quite≤ ∞ easy to show that the in Eq. (9) and Eq. (10) where r(z, τ) and r(x, z, τ) evolution equation in infinite space indeed reduces to the describes the approach towardsT the NESS solutionsC in skew-fractional equation in Eq. (3). A more direct and Eq. (11). It is easy to see that (x, z, τ) and (z, τ) detailed proof is given in [27]. Cr Tr 5

Conclusion: In this Letter, we have studied anoma- 112(4), 2014. lous transport in a one-dimensional system with two con- [11] Suman G. Das, Abhishek Dhar, Keiji Saito, Christian B. served quantities, in the open system setup. Starting Mendl, and Herbert Spohn. Numerical test of hydrody- from a microscopic description and acquiring knowledge namic fluctuation theory in the Fermi-Pasta-Ulam chain. Physical Review E - Statistical, Nonlinear, and Soft Mat- about scaling properties from numerical studies, we de- ter Physics, 90(1), 2014. rive exact expressions of the temperature profiles and the [12] Cédric Bernardin, Patrícia Gonçalves, Milton Jara, two point correlations in the steady state. We also study Makiko Sasada, and Marielle Simon. From Normal Dif- the evolution of these quantities towards steady state. fusion to Superdiffusion of Energy in the Evanescent Flip We explicitly show that the evolution of the temperature Noise Limit. Journal of Statistical Physics, 159(6):1327– profiles in this model is governed by a non-local operator 1368, 2015. defined inside a finite domain which correctly takes the [13] Cédric Bernardin and Gabriel Stoltz. Anomalous diffusion for a class of systems with two conserved quantities. previously obtained infinite system representation. We Nonlinearity, 25(4):1099–1133, 2012. provide numerical verifications of the analytical results. [14] Milton Jara, Tomasz Komorowski, and Stefano Olla. Su- Our work provides the first clear and transparent micro- perdiffusion of Energy in a Chain of Harmonic Oscillators scopic derivation of non-local heat equation describing with Noise. Communications in Mathematical Physics, anomalous transport in finite geometry and its connec- 339(2):407–453, 2015. tion to the corresponding skew-fractional equation in the [15] Herbert Spohn. Fluctuating hydrodynamics approach to infinite domain. equilibrium time correlations for anharmonic chains. Lec- ture Notes in Physics, 921:107–158, 2016. We thank C. Bernardin, C. Mejía-Monasterio, S. [16] Christian B. Mendl and Herbert Spohn. Current fluc- Olla and S. Lepri for very useful discussions. AK ac- tuations for anharmonic chains in thermal equilibrium. knowledges support from DST grant under project No. Journal of Statistical Mechanics: Theory and Experi- ECR/2017/000634. ment, 2015(3), 2015. [17] Giada Basile, Cédric Bernardin, and Stefano Olla. Mo- mentum conserving model with anomalous thermal conductivity in low dimensional systems. Physical Review Letters, 96(20), 2006. ∗ [email protected] [18] Cédric Bernardin, Patrícia Gonçalves, and Milton Jara. † [email protected] 3/4-Fractional Superdiffusion in a System of Harmonic ‡ [email protected] Oscillators Perturbed by a Conservative Noise. Archive § [email protected] for Rational Mechanics and Analysis, 220(2):505–542, [1] Onuttom Narayan and Sriram Ramaswamy. Anoma- 2016. lous Heat Conduction in One-Dimensional Momentum- [19] . Fractional Laplacian. Conserving Systems. Physical Review Letters, 89(20), [20] . Fractional Heat Equation. 2002. [21] G. M. Viswanathan, V. Afanasyev, Sergey V. Buldyrev, [2] Henk Van Beijeren. Exact results for anomalous trans- Shlomo Havlin, M. G.E. Da Luz, E. P. Raposo, and port in one-dimensional hamiltonian systems. Physical H. Eugene Stanley. Levy flights in random searches. Review Letters, 108(18), 2012. Physica A: Statistical Mechanics and its Applications, [3] Herbert Spohn. Nonlinear Fluctuating Hydrodynamics 282(1):1–12, 2000. for Anharmonic Chains. Journal of Statistical Physics, [22] A. Zoia, A. Rosso, and M. Kardar. Fractional Laplacian 154(5):1191–1227, feb 2014. in bounded domains. Physical Review E - Statistical, [4] Victor Lee, Chi Hsun Wu, Zong Xing Lou, Wei Li Lee, Nonlinear, and Soft Matter Physics, 76(2), 2007. and Chih Wei Chang. Divergent and Ultrahigh Ther- [23] S. Lepri, C. Mejía-Monasterio, and A. Politi. A stochastic mal Conductivity in Millimeter-Long Nanotubes. Physi- model of anomalous heat transport: Analytical solution cal Review Letters, 118(13), 2017. of the steady state. Journal of Physics A: Mathematical [5] Stefano Lepri, Roberto Livi, and Antonio Politi. Univer- and Theoretical, 42(2), 2009. sality of anomalous one-dimensional heat conductivity. [24] Luca Delfini, Stefano Lepri, Roberto Livi, and Anto- Physical Review E - Statistical Physics, Plasmas, Fluids, nio Politi. Nonequilibrium invariant measure under heat and Related Interdisciplinary Topics, 68(6), 2003. flow. Physical Review Letters, 101(12), 2008. [6] Abhishek Dhar. Heat transport in low-dimensional sys- [25] J. Cividini, A. Kundu, A. Miron, and D. Mukamel. Tem- tems. Advances in Physics, 57(5):457–537, 2008. perature profile and boundary conditions in an anoma- [7] Stefano Lepri, Roberto Livi, and Antonio Politi. Heat lous heat transport model. Journal of Statistical Mechan- transport in low dimensions: Introduction and phe- ics: Theory and Experiment, 2017(1), 2017. nomenology. Lecture Notes in Physics, 921:1–37, 2016. [26] Stefano Lepri, Carlos Mejía-Monasterio, and Antonio [8] Abhishek Dhar. Heat conduction in a one-dimensional Politi. Nonequilibrium dynamics of a stochastic model of gas of elastically colliding particles of unequal masses. anomalous heat transport. Journal of Physics A: Math- Physical Review Letters, 86(16):3554–3557, 2001. ematical and Theoretical, 43(6), 2010. [9] S. Lepri, R. Livi, and A. Politi. On the anomalous ther- [27] Supplementary Material . mal conductivity of one-dimensional lattices. Europhysics [28] Stefano Lepri and Antonio Politi. Density profiles in open Letters, 43(3):271–276, 1998. superdiffusive systems. Physical Review E - Statistical, [10] Sha Liu, Peter Hänggi, Nianbei Li, Jie Ren, and Baowen Nonlinear, and Soft Matter Physics, 83(3), 2011. Li. Anomalous heat diffusion. Physical Review Letters, 1

Supplementary Material: Anomalous heat equation in a system connected to thermal reservoirs Priyanka1, Aritra Kundu1, Abhishek Dhar1, Anupam Kundu1 1International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India

Abstract We here provide the details of calculations for the results presented in the main text.

DYNAMICAL OPERATORS AND DISCRETE EQUATIONS

The explicit expression of the deterministic part of the Fokker Planck (FP) equation given by operators l, b, deﬁned as L L

L 1 X− l = (V 0(ηi+1) V 0(ηi 1))∂ηi V 0(ηL 1)∂ηL + V 0(η2)∂η L − − − − 1 i=1 2 2 = λT ∂ λ∂ V 0(η ) + λT ∂ λ∂ V 0(η ), Lb ` η1 − η1 1 r ηL − ηL L where T` and Tr are the temperatures of the reservoirs on the left and right, respectively. The stochastic part ex is given as L

L 1 ! X− = γ P (~η ) P (~η) (S1) Lex i,i+1 − i=1 where ~ηi,i+1 denote the conﬁguration after the exchange of variable i with i + 1. The two point function deﬁned as, 2 Ci,j = ηiηj . From the above FP equation for V (η) = η /2, the dynamical equation Ci,j = ηiηj s can easily be writtenh as i h i

C˙ ij = Ci+1,j Ci 1,j + Ci,j+1 Ci,j 1 + γ[Ci 1,j + Ci+1,j + Ci,j 1 + Ci,j+1 4Ci,j], 1 < i, j < L (S2) − − − − − − − C˙ i,i+1 = Ti+1 Ci 1,i+1 + Ci,i+2 Ti + γ[Ci 1,i+1 + Ci,i+2 2Ci,i+1], 1 < i < L (S3) − − − − − T˙i = 2[Ci,i+1 Ci 1,i] + γ[Ti+1 + Ti 1 2Ti], 1 i L (S4) − − − − ≤ ≤ The dynamical equations at the boundaries are given by

1. for i = j = 1

T˙ = 2λT + 2C 2λT + γ[T T ] (S5) 1 ` 1,2 − 1 2 − 1 2. for i = j = L

T˙L = 2λTr 2CL 1,L 2λTL + γ[TL 1 TL] (S6) − − − − − 3.i = 1 and 1 < j < L

C˙ 1,j = C2,j λC1,j + C1,j+1 C1,j 1 + γ[C1,j 1 + C1,j+1 + C2,j 3C1,j] (S7) − − − − − 4.j = L and 1 < i < L

C˙ i,L = Ci+1,L Ci 1,L Ci,L 1 λCi,L + γ[Ci 1,L + Ci+1,L + Ci,L 1 3Ci,L] (S8) − − − − − − − − 5.i = 1 and j = L

C˙ 1,L = C2,L C1,L 1 2λC1,L + γ[C2,L + C1,L 1 2C1,L] (S9) − − − − − blue,orange,green,red=1000,2000,3000,4000 2

y = k/L y = k/L

AAAB7XicbVBNS8NAEJ34WetX1aOXxVbwFJNe1INQ8OLBQwVjC20om+2mXbLZhN2NEEJ/hBcPKl79P978N27bHLT1wcDjvRlm5gUpZ0o7zre1srq2vrFZ2apu7+zu7dcODh9VkklCPZLwRHYDrChngnqaaU67qaQ4DjjtBNHN1O88UalYIh50nlI/xiPBQkawNlKnkV9H53eNQa3u2M4MaJm4JalDifag9tUfJiSLqdCEY6V6rpNqv8BSM8LppNrPFE0xifCI9gwVOKbKL2bnTtCpUYYoTKQpodFM/T1R4FipPA5MZ4z1WC16U/E/r5fp8NIvmEgzTQWZLwozjnSCpr+jIZOUaJ4bgolk5lZExlhiok1CVROCu/jyMvGa9pXt3DfrLbtMowLHcAJn4MIFtOAW2uABgQie4RXerNR6sd6tj3nrilXOHMEfWJ8/V56OYQ== AAAB7XicbVBNS8NAEJ34WetX1aOXxVbwFJNe1INQ8OLBQwVjC20om+2mXbLZhN2NEEJ/hBcPKl79P978N27bHLT1wcDjvRlm5gUpZ0o7zre1srq2vrFZ2apu7+zu7dcODh9VkklCPZLwRHYDrChngnqaaU67qaQ4DjjtBNHN1O88UalYIh50nlI/xiPBQkawNlKnkV9H53eNQa3u2M4MaJm4JalDifag9tUfJiSLqdCEY6V6rpNqv8BSM8LppNrPFE0xifCI9gwVOKbKL2bnTtCpUYYoTKQpodFM/T1R4FipPA5MZ4z1WC16U/E/r5fp8NIvmEgzTQWZLwozjnSCpr+jIZOUaJ4bgolk5lZExlhiok1CVROCu/jyMvGa9pXt3DfrLbtMowLHcAJn4MIFtOAW2uABgQie4RXerNR6sd6tj3nrilXOHMEfWJ8/V56OYQ==

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0 -0.002 0 k k

-0.004 -0.0001

,k (a) ,k (b)

AAAB63icbVA9TwJBEJ3DL8Qv1NJmI5hgQ+5o1I7ExhITT0jgQvaWOdiwt3fZ3TMhhN9gY6HG1j9k579xgSsUfMkkL+/NZGZemAqujet+O4WNza3tneJuaW//4PCofHzyqJNMMfRZIhLVCalGwSX6hhuBnVQhjUOB7XB8O/fbT6g0T+SDmaQYxHQoecQZNVbyqzV6We2XK27dXYCsEy8nFcjR6pe/eoOEZTFKwwTVuuu5qQmmVBnOBM5KvUxjStmYDrFrqaQx6mC6OHZGLqwyIFGibElDFurviSmNtZ7Eoe2MqRnpVW8u/ud1MxNdB1Mu08ygZMtFUSaIScj8czLgCpkRE0soU9zeStiIKsqMzadkQ/BWX14nfqN+U3fvG5VmLU+jCGdwDjXw4AqacAct8IEBh2d4hTdHOi/Ou/OxbC04+cwp/IHz+QOl+41d AAAB63icbVA9TwJBEJ3DL8Qv1NJmI5hgQ+5o1I7ExhITT0jgQvaWPdiwt3fZnTMhhN9gY6HG1j9k579xgSsUfMkkL+/NZGZemEph0HW/ncLG5tb2TnG3tLd/cHhUPj55NEmmGfdZIhPdCanhUijuo0DJO6nmNA4lb4fj27nffuLaiEQ94CTlQUyHSkSCUbSSX62Fl9V+ueLW3QXIOvFyUoEcrX75qzdIWBZzhUxSY7qem2IwpRoFk3xW6mWGp5SN6ZB3LVU05iaYLo6dkQurDEiUaFsKyUL9PTGlsTGTOLSdMcWRWfXm4n9eN8PoOpgKlWbIFVsuijJJMCHzz8lAaM5QTiyhTAt7K2EjqilDm0/JhuCtvrxO/Eb9pu7eNyrNWp5GEc7gHGrgwRU04Q5a4AMDAc/wCm+Ocl6cd+dj2Vpw8plT+APn8wengI1e 0 -0.006 0 k k

+ -0.008 + -0.0002 k k -0.010 AAAB7XicbVBNS8NAEJ3Ur1q/qh69LLaCp7DpRT0IBS8eKxhbaEPZbDft0s0m7G7EEvojvHhQ8er/8ea/cdvmoK0PBh7vzTAzL0wF1wbjb6e0tr6xuVXeruzs7u0fVA+PHnSSKcp8mohEdUKimeCS+YYbwTqpYiQOBWuH45uZ335kSvNE3ptJyoKYDCWPOCXGSu3603XDxfV+tYZdPAdaJV5BalCg1a9+9QYJzWImDRVE666HUxPkRBlOBZtWeplmKaFjMmRdSyWJmQ7y+blTdGaVAYoSZUsaNFd/T+Qk1noSh7YzJmakl72Z+J/XzUx0GeRcpplhki4WRZlAJkGz39GAK0aNmFhCqOL2VkRHRBFqbEIVG4K3/PIq8RvulYvvGrWmW6RRhhM4hXPw4AKacAst8IHCGJ7hFd6c1Hlx3p2PRWvJKWaO4Q+czx/TPI4K x =2.0 AAAB7XicbVBNS8NAEJ34WetX1aOXxVbwFJIiqAeh4MVjBWMLbSib7aZdursJuxuxhP4ILx5UvPp/vPlv3LY5aOuDgcd7M8zMi1LOtPG8b2dldW19Y7O0Vd7e2d3brxwcPugkU4QGJOGJakdYU84kDQwznLZTRbGIOG1Fo5up33qkSrNE3ptxSkOBB5LFjGBjpVbt6frc9Wq9StVzvRnQMvELUoUCzV7lq9tPSCaoNIRjrTu+l5owx8owwumk3M00TTEZ4QHtWCqxoDrMZ+dO0KlV+ihOlC1p0Ez9PZFjofVYRLZTYDPUi95U/M/rZCa+DHMm08xQSeaL4owjk6Dp76jPFCWGjy3BRDF7KyJDrDAxNqGyDcFffHmZBHX3yvXu6tWGW6RRgmM4gTPw4QIacAtNCIDACJ7hFd6c1Hlx3p2PeeuKU8wcwR84nz/WSI4M x =4.0

LC LC -0.0003 -0.012 AAACAXicbVC7TsMwFHXKq5RXgAmxWLRISECUdAG2Sl0YGIpEaaU2ihzXaa04TrAdpCqKWPgVFgZArPwFG3+D22YAypHu1dE598q+x08Ylcq2v4zSwuLS8kp5tbK2vrG5ZW7v3Mo4FZi0ccxi0fWRJIxy0lZUMdJNBEGRz0jHD5sTv3NPhKQxv1HjhLgRGnIaUIyUljxzr9aXd0JlV3nTy8Lj0LNPwlPd85pnVm3LngLOE6cgVVCg5Zmf/UGM04hwhRmSsufYiXIzJBTFjOSVfipJgnCIhqSnKUcRkW42PSGHh1oZwCAWuriCU/XnRoYiKceRrycjpEbyrzcR//N6qQrO3YzyJFWE49lDQcqgiuEkDziggmDFxpogLKj+K8QjJBBWOrWKDsH5e/I8adetC8u+rlcbVpFGGeyDA3AEHHAGGuAStEAbYPAAnsALeDUejWfjzXifjZaMYmcX/ILx8Q33iZX8 AAACAXicbVC7TsMwFHXKq5RXgAmxWLRISECUdAG2Sl0YGIpEaaU2ihzXaa04TrAdpCqKWPgVFgZArPwFG3+D22YAypHu1dE598q+x08Ylcq2v4zSwuLS8kp5tbK2vrG5ZW7v3Mo4FZi0ccxi0fWRJIxy0lZUMdJNBEGRz0jHD5sTv3NPhKQxv1HjhLgRGnIaUIyUljxzr9aXd0JlV3nTy8Lj0LNPwlPd85pnVm3LngLOE6cgVVCg5Zmf/UGM04hwhRmSsufYiXIzJBTFjOSVfipJgnCIhqSnKUcRkW42PSGHh1oZwCAWuriCU/XnRoYiKceRrycjpEbyrzcR//N6qQrO3YzyJFWE49lDQcqgiuEkDziggmDFxpogLKj+K8QjJBBWOrWKDsH5e/I8adetC8u+rlcbVpFGGeyDA3AEHHAGGuAStEAbYPAAnsALeDUejWfjzXifjZaMYmcX/ILx8Q33iZX8 p p -0.014

x =2k/pL x =2k/pL

AAAB93icbVBNT8JAEN36ifhB1aOXjWDiqRYu6sGExIsHD5hYIYGGbJctbNhu6+7UiA2/xIsHNV79K978Ny7Qg4IvmeTlvZnMzAsSwTW47re1tLyyurZe2Chubm3vlOzdvTsdp4oyj8YiVq2AaCa4ZB5wEKyVKEaiQLBmMLyc+M0HpjSP5S2MEuZHpC95yCkBI3XtUuXxojY86eh7Bdn1uNK1y67jToEXSTUnZZSj0bW/Or2YphGTQAXRul11E/AzooBTwcbFTqpZQuiQ9FnbUEkipv1sevgYHxmlh8NYmZKAp+rviYxEWo+iwHRGBAZ63puI/3ntFMIzP+MySYFJOlsUpgJDjCcp4B5XjIIYGUKo4uZWTAdEEQomq6IJoTr/8iLxas65497UynUnT6OADtAhOkZVdIrq6Ao1kIcoStEzekVv1pP1Yr1bH7PWJSuf2Ud/YH3+ABPIkjE= AAAB93icbVBNT8JAEN36ifhB1aOXjWDiqRYu6sGExIsHD5hYIYGGbJctbNhu6+7UiA2/xIsHNV79K978Ny7Qg4IvmeTlvZnMzAsSwTW47re1tLyyurZe2Chubm3vlOzdvTsdp4oyj8YiVq2AaCa4ZB5wEKyVKEaiQLBmMLyc+M0HpjSP5S2MEuZHpC95yCkBI3XtUuXxojY86eh7Bdn1uNK1y67jToEXSTUnZZSj0bW/Or2YphGTQAXRul11E/AzooBTwcbFTqpZQuiQ9FnbUEkipv1sevgYHxmlh8NYmZKAp+rviYxEWo+iwHRGBAZ63puI/3ntFMIzP+MySYFJOlsUpgJDjCcp4B5XjIIYGUKo4uZWTAdEEQomq6IJoTr/8iLxas65497UynUnT6OADtAhOkZVdIrq6Ao1kIcoStEzekVv1pP1Yr1bH7PWJSuf2Ud/YH3+ABPIkjE= 0.00 0.00

k 0.5 1.0 1.5 2.0 2.5 3.0 3.5 k 0.5 1.0 1.5 2.0 2.5 3.0 0 -0.02 (c) 0 -0.02 (d)

AAAB63icbVA9TwJBEJ3DL8Qv1NJmI5hgQ+5o1I7ExhITT0jgQvaWOdiwt3fZ3TMhhN9gY6HG1j9k579xgSsUfMkkL+/NZGZemAqujet+O4WNza3tneJuaW//4PCofHzyqJNMMfRZIhLVCalGwSX6hhuBnVQhjUOB7XB8O/fbT6g0T+SDmaQYxHQoecQZNVbyqzV2We2XK27dXYCsEy8nFcjR6pe/eoOEZTFKwwTVuuu5qQmmVBnOBM5KvUxjStmYDrFrqaQx6mC6OHZGLqwyIFGibElDFurviSmNtZ7Eoe2MqRnpVW8u/ud1MxNdB1Mu08ygZMtFUSaIScj8czLgCpkRE0soU9zeStiIKsqMzadkQ/BWX14nfqN+U3fvG5VmLU+jCGdwDjXw4AqacAct8IEBh2d4hTdHOi/Ou/OxbC04+cwp/IHz+QOpBY1f AAAB63icbVBNT8JAEJ3iF+IX6tHLRjDBC2m5qDcSLx4xsUICDdlut7Bhu9vsbk1Iw2/w4kGNV/+QN/+NC/Sg4EsmeXlvJjPzwpQzbVz32yltbG5t75R3K3v7B4dH1eOTRy0zRahPJJeqF2JNORPUN8xw2ksVxUnIaTec3M797hNVmknxYKYpDRI8EixmBBsr+fVGdFkfVmtu010ArROvIDUo0BlWvwaRJFlChSEca9333NQEOVaGEU5nlUGmaYrJBI9o31KBE6qDfHHsDF1YJUKxVLaEQQv190SOE62nSWg7E2zGetWbi/95/czE10HORJoZKshyUZxxZCSaf44ipigxfGoJJorZWxEZY4WJsflUbAje6svrxG81b5rufavWbhRplOEMzqEBHlxBG+6gAz4QYPAMr/DmCOfFeXc+lq0lp5g5hT9wPn8AqoqNYA== k,k k,k -0.04 -0.04 + + 0 0 k k -0.06

-0.06 yAAAB7nicbVBNS8NAEJ3Ur1q/qh69LLZCTyEpiHoQCl48VjC20Iay2W7apZtN3N0IIfRPePGg4tXf481/47bNQVsfDDzem2FmXpBwprTjfFultfWNza3ydmVnd2//oHp49KDiVBLqkZjHshtgRTkT1NNMc9pNJMVRwGknmNzM/M4TlYrF4l5nCfUjPBIsZARrI3Xr2bVjN8/rg2rNsZ050CpxC1KDAu1B9as/jEkaUaEJx0r1XCfRfo6lZoTTaaWfKppgMsEj2jNU4IgqP5/fO0VnRhmiMJamhEZz9fdEjiOlsigwnRHWY7XszcT/vF6qw0s/ZyJJNRVksShMOdIxmj2PhkxSonlmCCaSmVsRGWOJiTYRVUwI7vLLq8Rr2le2c9estRpFGmU4gVNogAsX0IJbaIMHBDg8wyu8WY/Wi/VufSxaS1Yxcwx/YH3+AEmujkQ= =0.25 yAAAB7nicbVBNS8NAEJ3Ur1q/qh69LLZCTyEpiHoQCl48VjC20Iay2W7apZtN3N0IIfRPePGg4tXf481/47bNQVsfDDzem2FmXpBwprTjfFultfWNza3ydmVnd2//oHp49KDiVBLqkZjHshtgRTkT1NNMc9pNJMVRwGknmNzM/M4TlYrF4l5nCfUjPBIsZARrI3Xr2bVjnzv1QbXm2M4caJW4BalBgfag+tUfxiSNqNCEY6V6rpNoP8dSM8LptNJPFU0wmeAR7RkqcESVn8/vnaIzowxRGEtTQqO5+nsix5FSWRSYzgjrsVr2ZuJ/Xi/V4aWfM5GkmgqyWBSmHOkYzZ5HQyYp0TwzBBPJzK2IjLHERJuIKiYEd/nlVeI17SvbuWvWWo0ijTKcwCk0wIULaMEttMEDAhye4RXerEfrxXq3PhatJauYOYY/sD5/AEapjkI= =0.50 LC LC -0.08 -0.08 AAACAXicbVC7TsMwFHXKq5RXgAmxWLRISECUdAG2Sl0YGIpEaaU2ihzXaa04TrAdpCqKWPgVFgZArPwFG3+D22YAypHu1dE598q+x08Ylcq2v4zSwuLS8kp5tbK2vrG5ZW7v3Mo4FZi0ccxi0fWRJIxy0lZUMdJNBEGRz0jHD5sTv3NPhKQxv1HjhLgRGnIaUIyUljxzr9aXd0JlV3nTy0LPPg5PdD8N85pnVm3LngLOE6cgVVCg5Zmf/UGM04hwhRmSsufYiXIzJBTFjOSVfipJgnCIhqSnKUcRkW42PSGHh1oZwCAWuriCU/XnRoYiKceRrycjpEbyrzcR//N6qQrO3YzyJFWE49lDQcqgiuEkDziggmDFxpogLKj+K8QjJBBWOrWKDsH5e/I8adetC8u+rlcbVpFGGeyDA3AEHHAGGuAStEAbYPAAnsALeDUejWfjzXifjZaMYmcX/ILx8Q33aZX8 AAACAXicbVC7TsMwFHXKq5RXgAmxWLRISECUdAG2Sl0YGIpEaaU2ihzXaa04TrAdpCqKWPgVFgZArPwFG3+D22YAypHu1dE598q+x08Ylcq2v4zSwuLS8kp5tbK2vrG5ZW7v3Mo4FZi0ccxi0fWRJIxy0lZUMdJNBEGRz0jHD5sTv3NPhKQxv1HjhLgRGnIaUIyUljxzr9aXd0JlV3nTy0LPPg5PdD8N85pnVm3LngLOE6cgVVCg5Zmf/UGM04hwhRmSsufYiXIzJBTFjOSVfipJgnCIhqSnKUcRkW42PSGHh1oZwCAWuriCU/XnRoYiKceRrycjpEbyrzcR//N6qQrO3YzyJFWE49lDQcqgiuEkDziggmDFxpogLKj+K8QjJBBWOrWKDsH5e/I8adetC8u+rlcbVpFGGeyDA3AEHHAGGuAStEAbYPAAnsALeDUejWfjzXifjZaMYmcX/ILx8Q33aZX8 p p

0.19 0.008

025) 0.18 )

. L = 1000

⌧ (0 0.17 0.006 L = 1500 y, ss L = 1000 ( T = 2000 0.16 L = 1500 0.004 Lj ) 0.15 = 2000 ⌧ AAAB/HicbVDLSsNAFJ3UV62v+Ni5CbZCBQlJN+qu4MaFiwrGFppQJtNJO3bycOZGqKH6K25cqLj1Q9z5N07bLLT1wIXDOfdy7z1+wpkEy/rWCguLS8srxdXS2vrG5pa+vXMj41QQ6pCYx6LlY0k5i6gDDDhtJYLi0Oe06Q/Ox37zngrJ4ugahgn1QtyLWMAIBiV19L2KK+8EZJejx9vq8NgFnB5VOnrZMq0JjHli56SMcjQ6+pfbjUka0ggIx1K2bSsBL8MCGOF0VHJTSRNMBrhH24pGOKTSyybXj4xDpXSNIBaqIjAm6u+JDIdSDkNfdYYY+nLWG4v/ee0UglMvY1GSAo3IdFGQcgNiYxyF0WWCEuBDRTARTN1qkD4WmIAKrKRCsGdfnidOzTwzratauW7maRTRPjpAVWSjE1RHF6iBHETQA3pGr+hNe9JetHftY9pa0PKZXfQH2ucP/iCUZg== p , 0.14 (e) (f) 0.002 AAAB63icbVA9TwJBEJ3DL8Qv1NJmI5hgQ+5o1I7ExhITT0jgQvaWOdiwt3fZ3TMhhN9gY6HG1j9k579xgSsUfMkkL+/NZGZemAqujet+O4WNza3tneJuaW//4PCofHzyqJNMMfRZIhLVCalGwSX6hhuBnVQhjUOB7XB8O/fbT6g0T+SDmaQYxHQoecQZNVbyqzW8rPbLFbfuLkDWiZeTCuRo9ctfvUHCshilYYJq3fXc1ARTqgxnAmelXqYxpWxMh9i1VNIYdTBdHDsjF1YZkChRtqQhC/X3xJTGWk/i0HbG1Iz0qjcX//O6mYmugymXaWZQsuWiKBPEJGT+ORlwhcyIiSWUKW5vJWxEFWXG5lOyIXirL68Tv1G/qbv3jUqzlqdRhDM4hxp4cAVNuIMW+MCAwzO8wpsjnRfn3flYthacfOYU/sD5/AGsD41h AAAB63icbVA9TwJBEJ3DL8Qv1NJmI5hgQ+5o1I7ExhITT0jgQvaWOdiwt3fZ3TMhhN9gY6HG1j9k579xgSsUfMkkL+/NZGZemAqujet+O4WNza3tneJuaW//4PCofHzyqJNMMfRZIhLVCalGwSX6hhuBnVQhjUOB7XB8O/fbT6g0T+SDmaQYxHQoecQZNVbyq7XostovV9y6uwBZJ15OKpCj1S9/9QYJy2KUhgmqdddzUxNMqTKcCZyVepnGlLIxHWLXUklj1MF0ceyMXFhlQKJE2ZKGLNTfE1Maaz2JQ9sZUzPSq95c/M/rZia6DqZcpplByZaLokwQk5D552TAFTIjJpZQpri9lbARVZQZm0/JhuCtvrxO/Eb9pu7eNyrNWp5GEc7gHGrgwRU04Q5a4AMDDs/wCm+OdF6cd+dj2Vpw8plT+APn8wetlI1i 025 . 0.13 (0 0.000 T AAACGXicbZDLSsNAFIYn9VbrLerSzWArtKAhKYi6K7hxWaGxhSaEyXTSDp1cmJkIJfQ53PgqblyouNSVb+OkzaK2Hhj4+P9zmHN+P2FUSNP80Upr6xubW+Xtys7u3v6Bfnj0IOKUY2LjmMW85yNBGI2ILalkpJdwgkKfka4/vs397iPhgsZRR04S4oZoGNGAYiSV5OlWzQmRHGHEss60bhpm8/LckShtXCzoXiZEYTZqnl5VNCu4ClYBVVBU29O/nEGM05BEEjMkRN8yE+lmiEuKGZlWnFSQBOExGpK+wgiFRLjZ7LQpPFPKAAYxVy+ScKYuTmQoFGIS+qoz31cse7n4n9dPZXDtZjRKUkkiPP8oSBmUMcxzggPKCZZsogBhTtWuEI8QR1iqNCsqBGv55FWwm8aNYd43qy2jSKMMTsApqAMLXIEWuANtYAMMnsALeAPv2rP2qn1on/PWklbMHIM/pX3/AuavnmY= 0.1 0.20.3 0.4 0.5 0.05 0.10 0.15 0.20 0.25 0.30 ⌧ = t/L3/2 ⌧ = t/L3/2

AAAB+HicbVBNT8JAEN36ifhV9ehlI5h4KgUP6sGExIsHD5hYIYFKtssWNmy3ze6UhDT8Ey8e1Hj1p3jz37hADwq+ZJKX92YyMy9IBNfgut/Wyura+sZmYau4vbO7t28fHD7qOFWUeTQWsWoFRDPBJfOAg2CtRDESBYI1g+HN1G+OmNI8lg8wTpgfkb7kIacEjNS17XIHSHoNlbun7LxSm5S7dsl13BnwMqnmpIRyNLr2V6cX0zRiEqggWrerbgJ+RhRwKtik2Ek1Swgdkj5rGypJxLSfzS6f4FOj9HAYK1MS8Ez9PZGRSOtxFJjOiMBAL3pT8T+vnUJ46WdcJikwSeeLwlRgiPE0BtzjilEQY0MIVdzciumAKELBhFU0IVQXX14mXs25ctz7Wqnu5GkU0DE6QWeoii5QHd2iBvIQRSP0jF7Rm5VZL9a79TFvXbHymSP0B9bnD/cakgw= AAAB+HicbVBNT8JAEN36ifhV9ehlI5h4KgUP6sGExIsHD5hYIYFKtssWNmy3ze6UhDT8Ey8e1Hj1p3jz37hADwq+ZJKX92YyMy9IBNfgut/Wyura+sZmYau4vbO7t28fHD7qOFWUeTQWsWoFRDPBJfOAg2CtRDESBYI1g+HN1G+OmNI8lg8wTpgfkb7kIacEjNS17XIHSHoNlbun7LxSm5S7dsl13BnwMqnmpIRyNLr2V6cX0zRiEqggWrerbgJ+RhRwKtik2Ek1Swgdkj5rGypJxLSfzS6f4FOj9HAYK1MS8Ez9PZGRSOtxFJjOiMBAL3pT8T+vnUJ46WdcJikwSeeLwlRgiPE0BtzjilEQY0MIVdzciumAKELBhFU0IVQXX14mXs25ctz7Wqnu5GkU0DE6QWeoii5QHd2iBvIQRSP0jF7Rm5VZL9a79TFvXbHymSP0B9bnD/cakgw=

FIG. S1. Data collapse of the correlation functions and temperature profile confirming the scaling behaviors in Eqs. (S10) and (S11). Figures (a) and (b) show the data collapse as a function of the scaling variable y = (i + j)/2L with four systems√ sizes L = 1000 (Blue), L = 2000 (Orange), L = 3000 (Green) and L = 4000 (Red), for two fixed values of x = |j − i|/ L. Figures (c) and (d), show the data collapse as a function of the scaling variable x with the above four system sizes for two fixed values of y. The collapse are so good that other colors not visible. Figure (e) describes the scaling behavior for the evolution of the temperature T (y, τ) = T (τL3/2) at a fixed position y = 0.025 for different system sizes as a function of the scaled time byLc √ 3/2 τ = t/L . Note that, the temperatures are of O(1) whereas the correlations are of O(1/ L). Also note√ from figures (c) and (b) that Css(x → ∞, z) = 0. Last figure (d) establishes that the current in the system is of order 1/ L and also evolves in 3/2 scaled time τ = t/L . The other parameters in the simulation are γ = Λ = 1, T` = 1.1,Tr = 0.9.

DERIVATION OF CONTINUUM EQUATIONS FROM DISCRETE EQUATIONS

Now, we want to get a continuum description of the bulk discrete dynamical equations derived in the previous 2 section. In the non-stationary regime, we numerically find that the temperature profile Ti(t) = ηi(t) and the two-point correlations C (t) = η (t)η (t) for i = j have the following scaling forms h i i,j h i j i 6 1 i j i + j t Ci,j(t) = | − |, , , (S10) √LC √L 2L L3/2 i t T (t) = , , (S11) i T L L3/2 3 in the leading order for large L which are also supported by numerical evidence shown in Fig S1. In figs. (S1a), (S1b, (S1c) and (S1d), verify the scaling behaviors of the correlations in Eq. (S10). Figures (S1c) and (S1d), describes i j i+j t scaling behavior with respect to time. Using these, we define continuum ordinates as, | − | = x, | | = y, = τ √L 2L L3/2 and 1 = , where x (0, ) and y (0, 1). In the following, we insert this scaling form and Taylor expand in √L ∈ ∞ ∈ = 1/√L. Keeping terms to leading order in we obtain the continuum equations. 1 Bulk Equations, i j 2 We start with the discrete| − | ≥ equation in bulk:

C˙ i,j = (Ci 1,j Ci+1,j + Ci,j 1 Ci,j+1 γ[Ci,j 1 + Ci,j+1 + Ci+1,j + Ci 1,j 4Ci,j]) , (S12) − − − − − − − − − using above scaling deﬁnitions, we can write the above mentioned discrete equation as, 2 2 2 2 4∂ (x, y, τ) = (x , y ) (x + , y + ) + (x + , y ) (x , y + ) τ C − C − − 2 − C 2 C − 2 − C − 2 2 2 2 2 γ (x + , y ) + (x , y + ) + (x + , y + ) + (x , y ) 4 (x, y) , (S13) − C − 2 C − 2 C 2 C − − 2 − C which by Taylor expansion of each terms in x, y and τ, we obtain the leading order terms for continuum dynamical equation as 4∂ (x, y, τ) = 23∂ (x, y, τ) + 2γ3∂2 (x, y, τ) . (S14) τ C yC xC At the dominant order (o(3)), we ﬁnd, ∂ (x, y, τ) + γ∂2 (x, y, τ) = 0 (S15) yC xC 2 Nearest neighbor term, j = i + 1

C˙ i,i+1 = Ti+1 Ci 1,i+1 + Ci,i+2 Ti + γ [Ci 1,i+1 + Ci,i+2 2Ci,i+1] (S16) − − − − − after proper scaling, we get, 2 2 4∂ (, y + , τ) = (y + 2) (y) (2, y + 2) + (2, y) + γ (2, y) + (2, y + 2) 2 (, y + ) . τ C 2 T − T − C C C C − C 2 (S17) Expanding above equation in x and y, and keeping the relevant order terms in we get the continuum equation as 4∂ (0, y, τ) = 2 (∂ (y, τ) + 2γ∂ (0, y, τ)) + O(3) (S18) τ C yT xC and hence to the dominating order, the governing continuum equation is ∂ (y, τ) + 2γ∂ (0, y, τ) = 0 (S19) yT xC 3 Diagonal term i = j Next is the diagonal term where i = j,

T˙i = 2[Ci,i+1 Ci 1,i] + γ[Ti+1 + Ti 1 2Ti]. (S20) − − − − this in continuum limit given as 2 2 3∂ (y, τ) = 2 (, y + ) (, y ) + γ (y + 2) + (y 2) 2 (y) (S21) τ T C 2 − C − 2 T T − − T After expansion, we arrive at h γ i 3∂ (y, τ) = 2 2∂ (0, y, τ) + 3 ∂2 (y, τ) + O(4) (S22) τ T yC 4 y T Hence, the leading order term is ∂ (y, τ) = 2∂ (0, y, τ) (S23) τ T yC 4

4 Current The microscopic energy current in the system is deﬁned through

2 ∂t ηi = [ji i+1 ji 1 i], (S24) h i − → − − →

where ji i+1 = 2Ci,i+1 γ(Ti+1 Ti). The stochastic part of the current decays as O(1/L) and in the macroscopic→ limit− goes to zero.− In the− continuum limit, the deterministic part contributes in the leading order to provide, j = 2 (0, y, τ)/√L. − C

SOLUTION IN THE STEADY STATE AS WELL AS IN THE RELAXATION REGIME

The above analysis gives us the following bulk equations for the system,

∂ (x, y, τ) = γ∂2 (x, y, τ) , (S25) yC − xC ∂ (y, τ) = 2γ∂ (x, y, τ) (S26) yT − xC x=0 ∂ (y, τ) = 2∂ (0, y, τ) (S27) τ T yC Solutions of these equations for (x, y, τ) and (y, τ) have two parts. One part describes the steady state and the other part describing the relaxationC to the steadyT state:

(y, τ) = (y) + (1 y, τ) (S28) T Tss Tr − (x, y, τ) = (x, y) + (x, 1 y, τ). (S29) C Css Cr − It is easy to show that (x, z) and (z) satisfy Css Tss ∂ (x, z) = γ∂2 (x, z) , (S30) zCss xCss ∂ (z) = 2γ∂ (x, z) . (S31) zTss xCss x=0 under the transformation of z = 1 y with boundary conditions ss(x , z) = 0, ss(0) = T` and ss(1) = Tr. On the other hand, the relaxation parts− satisfy C → ∞ T T

∂ (x, z, τ) = γ∂2 (x, z, τ) , (S32) zCr xCr ∂ (z, τ) = 2γ∂ (x, z, τ) (S33) zTr xCr x=0 ∂ (y, τ) = 2∂ (0, z, τ). (S34) τ Tr − zCr with appropriate boundary conditions speciﬁed in Sec..

Stationary state solution

In stationary state, we ﬁrst solve Eq. (S30) and Eq. (S31). We want to solve these equations along with the boundary conditions

(i) (x, z 0) = 0, (ii) (x , z) = 0, (iii) (x = 0, z) = d. (S35) Css → Css → ∞ Css The initial condition in (i) is easy to understand. The boundary condition (ii) is obtained from our numerical observation [see fig. S1(c,d) ]. Where the last boundary condition (iii) is obtained by observing that LHS of (S34) is zero in steady state; hence (∂z ss(0, z) = 0). The unknown constant d will be fixed by the temperatures at the boundary. The first equation isC easy to solve by taking Laplace transform in z along with boundary conditions. Finally inverting the Laplace transform, we find, the solution is given by,

x (x, z) = d erfc (S36) Css √4γz 5

2 R x t2 where, erfc is the complimentary error function deﬁned as erfc(x) = 1 dte− . Now, using this solution in the − √π 0 Eq. S31, we get r 4γ ∂ (z) = d zTss − πz r 4γz (z) = (0) 2d (S37) Tss Tss − π

The constants ss(0) and d will now be determined from the boundary conditions of temperature field, (z = 0) = Tr and (z = 1) =T T ,. We finally have, T T ` r 4γ T T = 2d ` − r − π ∆T rπ d = , (S38) − 4 γ where ∆T = (T` Tr) is the temperature difference between the left and right heat baths. Reverting now back to y variables using z −= 1 y, the exact expressions for the steady state temperature profile and correlations are, − p ss(y) =Tr + ∆T 1 y, (S39) T − " # ∆T rπ x ss(x, y) = erfc p (S40) C − 4 γ 4γ(1 y) − (S41)

ss Hence the current in the system is given by j = J , where, ss √L r 2 ss(0, y) ∆T π 1 jss = C = (S42) − √L 2 γ √L

Solution in the relaxation regime

Here we solve Eqs. (S32), (S33) and (S34) (Note that these are also Eqs. (16),(17) and (18) of the main text). For convenience, we rewrite these equations here,

∂ (x, z, τ) = γ∂2 (x, z, τ), (S43) zCr xCr ∂ (z, τ) = 2γ [∂ (x, z, τ)] , (S44) zTr xCr x=0 ∂ (z, τ) = 2∂ (0, z, τ). (S45) τ Tr − zCr

We want to solve (S43) with boundary conditions r(x, z) x = 0 and (S44). The Greens function g(x, z) of this γ 2 C | →∞ equation with above BC’s satisﬁes, ∂zg(x, z) = 2 ∂xg(x, z), where, g(x, z) is given by g(x, z) = √4γzh(x/√4γz) where, −w2 h(w) = e w erfc(w), hence, the general time dependent solution is written as, π − Z z (x x0)2/(4γz) e− − 1 R z 2 r(x, z, τ) = 2 dx0 r(x0, 0, τ) 2γ 0 dz0(g(x, z z0)∂z0 r(z0, τ)) C 0 √4πγz C − − T 1 ∂z0 r(z0, τ)g(x, z) z0 0 (S46) − 2γ T | →

With the initial condition r(x, 0, τ) = 0 the ﬁrst term drops out. It is easy to check that the remaining part satisﬁes (16) with boundary conditionC (17) as follows,

Z z 1 2 ∂x r(x, z, τ) x 0 = dz0∂xg(x, z z0))∂z0 r(z0, τ) + ∂xg(x, z))∂z0 r(z0, τ) z0 0 C | → − 2γ 0 − T T | → Z z 1 2 1 = dz0∂z0 r(z0, τ) + ∂z0 r(z0, τ) z0 0 = ∂z r(z, τ) (S47) 2γ 0 T T | → 2γ T 6 where we have used ∂xg(x, z) x 0 = 1. Further using the fact that g(x, z z0)∂z0 Tr(z0) z0 z 0 we can simplify (S46) as, | → − − | → →

Z z 1 2 r(x, z, τ) = dz0g(x, z z0)∂z0 r(z0, τ) + ∂z0 r(z0, τ)g(x, z) z0 0 g(x, z z0)∂z0 r(z0, τ) z0 z C − 2γ 0 − T T | → − − T | → 1 Z z = dz0∂z0 (g(x, z z0))∂z0 r(z0, τ) 2γ 0 − T Z z x2/(4γ(z z0)) 1 e− − = dz0 p ∂z0 r(z0, τ) (S48) − √γ 0 4π(z z ) T − 0 which gives the relaxation of the correlation fields. The evolution of temperature field is obtained from (18) by putting x 0 in the above expression for (x, z, τ), we immediately have, → Cr Z z ∂z0 r(z0, τ) 1 ∂τ r(z, τ) = κ ∂z T dz0, 0 z 1, where, κ = . (S49) T √z z ≤ ≤ √πγ 0 − 0 For infinite system, one can perform the same procedure as done above for finite system using appropriate control parameter and obtain

Z z ∂z0 r(z0, τ) ∂τ r(z, τ) = κ ∂z T dz0, z . (S50) T √z z0 − ∞ ≤ ≤ ∞ −∞ −

Fractional evolution of temperature in an inﬁnite line

In the previous section we have derived an equation for evolution of temperature the field in an finite system where y (0, 1). One can extend the same calculation in an finite system of length L and obtain the same set of bulk equations∈ which now hold for y [0,L]. We are interested in the behavior of the evolution of temperature profile in L limit, where the effect of∈ boundaries are not important. The evolution equations for the relaxation parts in this→ case ∞ are,

γ∂2 (x, y, τ) = ∂ (x, y, τ) xCr − yCr ∂y r(y, τ) = 2γ∂x r(x, y, τ) x 0 (S51) T − C | → ∂ (y, τ) = 2∂ (0, y, τ) τ Tr yCr where 0 x and 0 y L. To proceed, we introduce the orthonormal and complete basis in y [0,L] , ≤1 ≤inπy/L ∞ ≤ ≤ ∈ φ±(y) = e± for n 1 and φ (y) = 1/√L. Expanding the correlations and temperature in this basis as n √L ≥ 0 Fourier series we get,

X + + (x, y, τ) = Aˆ (x, τ) + Aˆ (x, τ)φ (y) + Aˆ−(x, τ)φ−(y) Cr 0 n n n n n=1 X + + (y, τ) = Tˆ (τ) + Tˆ (τ)φ (y) + Tˆ−(τ)φ−(y) (S52) T 0 n n n n n=1

R L R L R L where Aˆ±(x, τ) = (x, y, τ)φ±(y)dy, Aˆ (x, τ) = (x, y, τ)φ dy, Tˆ±(τ) = (y, τ)φ±(y)dy, Tˆ (τ) = n 0 Cr n 0 0 Cr 0 n 0 Tr n 0 R L (y, τ)φ dy. These gives the following diﬀerential equations for the components, 0 Tr 0 2 2 2 ∂ Aˆ±(x, τ) =((1 i)α ) Aˆ±(x, τ), ∂ Aˆ (x, τ) = 0 (S53) x n ∓ n n x 0 inπ ˆ ˆ Tn±(τ) = 2γ∂xAn±(x, τ) x 0. (S54) ± L − | →

q nπ where αn = 2Lγ . The solution to these equations are in general given as,

ˆ αn(1 i)x ˆ An±(x, τ) =an±(τ)e± ∓ , A0(x, τ) = d(τ)x + e(τ) 7

We choose solutions which do not blow up at inﬁnity at large x and obey the boundary conditions. We have,

∞ X αn(1+i)x (x, y, τ) =e(τ) + a−e− φ−(y) + c.c. (S55) Cr n n n=1 where c.c. stands for complex conjugate. e(τ) is zero because there is no time-dependent source in the system. Using above equations, , we get,

(1 i) ˙ nπ Tˆ± = 2γa∓(τ)α ∓ , Tˆ∓ = 2i a±(τ) (S56) n n n (nπ/L) n ∓ L n ˙ Upon combining these two, we get Tˆ0 = 0 and

˙ 1 3/4 Tˆ∓ = (1 i)λ Tˆ∓, n = 1, 2, 3... (S57) n −√2γ ± n n where λ = (nπ/L)2. This can be interpreted in domain y [0,L] as, n ∈ 1 3/4 1/4 1 ∂τ r(y, τ) = ( ∆ ∆ ) r(y, τ) = L r(y, τ) (S58) T −√2γ | | − ∇| | T −√2γ T

3/4 where L is an positive operator defined by its action as, Lφn±(y) = λn (1 i sgn(n))φn±(y). With L the spectrum becomes continuous as well as the eigenfunctions become plane wave.− Thus in infinite system at→ equilibrium, ∞ the evolution of temperature profile is given by a skew-symmetric fractional Laplacian given in Eq.(3) of the main text. One can alternatively see this equivalence from the integro-differential evolution in infinite space given in Eq. (S50). Let us write this equation as Z y 1 ∂y0 f(y0) ∂τ r(y, τ) = r(y, τ), where, f(y) = ∂y dy0. (S59) T L∞T L∞ √πγ √y y0 −∞ − Using the identity

Z y 1 √π dz eiqz = eiqy √y z √iq −∞ − one can easily show that r iqy iqy 1 3/2 e = λq e , λq = [1 i sgn(q)] q L∞ 2γ − | | which is same as the Fourier spectrum of the skew-symmetric fractional Laplacian given in Eq.(3) of the main text.

Series solution of the Fractional PDE in ﬁnite domain

The evolution of the relaxation part of the temperature proﬁle i.e. (1 y, τ) = (y, τ) (y) is given by Tr − T − Tss Eq. (S49). Note that, r(z, τ) is zero at both the boundaries: z = 0 and z = 1. As a result it natural to expand T P ˆ this function in αn(z) = √2 sin(nπz), n = 1, 2, 3... complete basis deﬁned in z (0, 1), as r(z, τ) = n θn(τ)αn(z). Substituting this form in Eq. (S49), we have, ∈ T Z z X ˙ X φn(z0) θˆnαn(z) = κ θˆn(τ)(nπ)∂z dz0 (S60) √z z n n 0 − 0 0 Now let us expand the function f (z) = ∂ R z φn(z ) dz also in orthogonal basis α (y), n = 1, 2..... Let the expansion n z 0 √z z0 0 n P R 1− is given as fn(z) = l=1 ζnlαl(z) where ζnl = 0 dz fn(z) αl(z). As a result we have, X ˙ X θˆnαn(z) = κ θˆn(τ)(nπ)ζnlαl(z) (S61) n=1 n,l=1 8

◆●■ ◆●■◆●■◆●■◆●■◆●■◆●■◆●■ ◆●■◆●■◆●■◆●■◆●■◆●■ ◆●◆■●◆■●◆■●◆●■◆●■◆●■ ◆●■◆●◆■●◆■●◆■●◆■●◆■●■ ◆●■◆●■◆●■◆●■◆●■◆●■ 4 ●◆■●◆■●◆■●◆■●◆■ 10 ◆●■◆●■◆●■◆●■◆ ●◆●◆●■◆●■◆●■◆●■◆●■■ ●◆●■◆●■◆●■◆●■◆●■◆■■ ◆●◆●■◆●■◆●■◆●■◆■ ●◆●■◆●■◆●■◆●■■ ●◆●■◆●■◆●■◆■ ●◆●◆●■◆●■◆■ ●◆●◆●■◆■■ ●◆●■◆■■ ●◆●◆●■◆■ 1000 ●◆●◆■■ ●◆●◆■■ n ●◆■■ ● Re(μ ) ● ◆●◆■ n μ ● ◆ ■ ● ◆ ■ ● ◆ ■ ■ Im(μn) ● ◆ ■ 100 ◆ ■ ● π 3/2 ◆ ◆ (nπ) ■ 2

● 10 ◆ 1 5 10 50 100 n

FIG. S2. The real and imaginary part of the alternate eigenvalues for the matrix B. The ﬁrst 4 eigenvalues are completely real p π 3/2 and distinct. The higher eigenvalue comes in pairs of µn(1 ± i). For large n, the eigenvalues are close to 2 (nπ) (1 ± i). For smaller n, there is a deviation from asymptomatic scaling.

Using orthogonality, this can be written in vector notation as (θˆ = n θˆ ), n h | i ˙ θˆ = κ B θˆ , (S62) | i | i 1 where Bnk = (nπ)ζnk. If R is the matrix which diagonalizes B as R− BR = Λ, then the time dependent solution is κΛτ 1 given as θˆ(τ) = Re R− θˆ(0) and temperature at time τ is given as (y, τ) = ss(y) + α(1 y) θˆ(τ) . The eigenvalues| i of the bounded| skew-fractionali laplacian, B have interesting behavior,T theT first fourh of them− | are reali and distinct. The higher eigenvalues all come in complex conjugate pairs of form µn(1 i). For large n, µn goes as p π 3/2 ± 2 (nπ) , but for smaller n there is a systematic deviation due to the effect of finite domain. In Fig. S2, the real p π 3/2 part of alternate eigenvalues are plotted as a function of n, where the asymptotic scaling with 2 (nπ) is seen P 1 clearly for large n. The eigenvectors of the operator is defined as ψn(y) = l=1 Rnl− αl(y). Numerically computing this gives, the first six eigenvectors to be completely real. The eigenvectors corresponding to higher eigenvalues are complex and and comes in pairs. The real and imaginary parts of the first few eigenvectors are shown in Fig.S3(a). In Fig.S3(b), the real and the imaginary part of the eigenvector for n 7 is plotted in polar plots. For plane wave solutions these would have been circles of length 1, here the polar plot≥ shows a spiral decay to origin owing to the skewness of the operator. As the temperature field evolves at much faster timescales compared to the correlation field, the time dependent solution for correlations r(x, 1 y, τ) is governed by the evolution of the temperature field. The solution for evolution of correlations is writtenC as, −(x, y, τ) = (x, 1 y, τ) + (x, y) , where, C Cr − Css Z z x2/(4γ(z z0)) e− − (x, z, τ) = dz0 p ∂z0 Tr(z0, τ) C − 0 4πγ(z z ) − 0 Z z x2/(4γ(z z0)) X ˆ e− − = θn(τ)(nπ) dz0 p φn(z0) (S63) 0 4πγ(z z ) n=1 − 0 where, φn(y) = √2 cos(nπy), n 1 and φ0(y) = 1. The integral can be evaluated explicitely and doing the summations gives the evolution of≥ the correlation fields. 9

ψ2(x) ψ1(x) ψ3(x) ψ4(x) ●●●●●●●● ●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■● 2 ●●●●●● ●●●● 10 ●●●●●●●●●● ●●●●● 0.0 ●● ●●●●● ●●● ●●● ●● ●●● ●● ●● ●●●● ●● ●●● ●● ●● ● ●● ●●●● ●● ●● ● ●● ● ●● ●●●● ● ●●● ● ●●● ● ●● ●●●● ● ●●● ● ●● ● ● ●●●●●● ● ●●●● ●● ●● 0 ●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●● ■■■■■■■● ●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●●●●●●●●●●●●● ●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■● ■■■■■■■■■■■● ●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●●●● ● ■■■■■■■■■■■■■■■● ●● ● 0 ●●●●●●●●●●●●●●●●● ● 0 ●●●●●●●●●●●●●●●●●● ● ● ● ● 0.5 ●● ● - ●● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● -2 ● -10 ● ●● ● ● ● ● ● ● ●● ● ● -50 -1.0 ●● ● ● ● ● ● ● ● ●● -4 -20 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● -1.5 ● ● 6 ● ● -30 ● -100 ●● ● - ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ●●● ● -8 ● -40 ●●● ●●● ● ● ● ● ● ● -2.0 ●●●●●● ●●●●● ●● ● ● ● ●●● ●● ●●●●●● ●●● -150 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x x x

ψ5(x) ψ6(x) ψ7(x) ψ8(x) ■ ■ 50 ■■■■■ 200 ■ ■ 50 ■■■ ●● 150 ■ ■ ■ ■■ ■ ■ ■●● ● ■■ ●●●● ■ ■ ■ ●●■ ●● ■ ●■ ● ■ ■ ■ ● ■ ● ■ ■ ● ■ ● ■ ■ ● ■■ ●● ■ ● ■ ● ■ ■■■■■■■■■●■●■●■●■●●●●●●●●●●● ■ ● ● ■ ●●● ■ ●■●■●■■●●■●■●■●■●●●■■■●■●■●■●■●■●■●■■■●●●■●■●■●■●■●■●■●●■■●■●●●●●●●●● ■■■■■■ ●●● ■■ ● ■ ●■ 150 ■■■■■●●● ●●● ● ■ ● 0 ■■■■■■ ●●■●■■■ ●● ● ●●●●●● ■■■■ ●●■■■ ● ■ ■ ■■■ ●●●●● ●●●● ■ ■ ●■●■●■●■●■●●●■■■●■●■●■●■●■●■●■●■●■●■●■●■●■●■●■●■●■●■●■■●●■●■●■■■■■■■●■●■●●●●■■■■■ ●●● ■■ ● ■ ● ●■ 100 ●● ● ■ 0 ■■■■●●●●●●●●● ■ ● ■ ● ■ ● ■■ ●■ ● ■ ■ ■ ■■■■■●■● ● ● ● ●● ●● ■ ■ ● ● 100 ■ ●●● ■ ● -50 ● ● ●●● ●● ■ ■ ■ 50 ■ 50 ●● - ●● ● ■ ■ 50 ● ■● -100 ●●●●■ ■ ● ■ ●● ●● ● ● ● ■■■■■■ ● ■ ●● ■ ● ●●●●●●●■ ■ ■ ● ■ ■■■■■■■■■■■ ● ■ ●● ■■● ■ ● ●●●●●●●●●●■■●■●■●■●■●■■■■■■■■■■ ●●●●●●●●●●■●■●■●■●■●●●●● ■■ ● ■ ● ●●●●●●●■●■●■●■■■■■■■■ ●●●●●●●■●■●■●■●■●■●■■■■■■■ ●● ■ ● ■ 0 ●■■■■■■■■■ ●●●●●●●●●●■●■●●■■●■●■●■●■●■●■●■■■■■■■■■■ ●●● ■■ ●● ● ●■ -100 ■ 0 ●■■■■■■■ ●●●●●●●●■●■●■●■●■●■●■●■●■■■■■■■ ●●●●●■■■■●●●● ■■ ● ■ ● ● ●■ ●●●● ■ ●● ■ ●●●●●■■■■■■■■■ ● ● ■ -150 ■ ●●●●●■●■●● ● ■ ● ■ ■ ■ ■ ●● ● ● ■ ■ ● ● ■● ■ ■■ ■ ●●●●●■ ■ ■■ ● ■ ■ ■ ● ■ ■ ■■■■■ ● ● ■ ● ■ ■ -50 ● ■ ■ ■ ■ ●● -200 ■ ●●● -150 ■ -50 ■■■■ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x x x

ψ7(x) ψ9(x) ψ11(x) ψ13(x) ● 50 ● ● ● ● ● ● ● ● ● ● ● ● ● 50 50 ● ● ● ● ● ● ● ● ● ● 100 ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●● ● ● ● ● ● ●● ●●●●●●●●●● ●● ● ● ● ● 0 ● ●●●●●●●●● ● ● ●●● ● ● ● ● ● ● ● ● ●●●●●● ●● ● ● ● ● ● ● ● ● ●● ●●●●●●●● ● ● ● ●● ●●●●●●●● ● ● ● ● ● 0 ● ● ●●●●●● ● ● ● ● ●●●●●●●●●●● ● ● ● ● ● ● ● 0 ● ●● ●●● ● ● ● ● ● ● ● ● (x)) (x)) (x)) ● (x)) ● ● 50 ● ● ● ● n n n ● n ● ● ● -50 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● -50 ●●●● Im(ψ ● -50 Im(ψ ● ●● ●● Im(ψ Im(ψ ● ● ●●●●●●●●●● ● ● ● 0 ● ● ●●●●●●●● ● ● -100 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● -100 ● ● ● ● -150 ● ● -50 ● ● ● ● ● -100 ● -40 -20 0 20 40 -60 -40 -20 0 20 40 -40 -20 0 20 40 60 -40 -20 0 20 40 60

Re(ψn(x)) Re(ψn(x)) Re(ψn(x)) Re(ψn(x))

ψ15(x) ψ17(x) ψ19(x) ψ21(x) ● ● ● ● 80 ● 50 ● ● 80 ● 40 ● ● ● ● ● ● ● ● ● 60 ● ● 60 ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 40 ● ● ●●●●●●●●●●●●● ● ● 40 ● ● ●●●●● ● 0 ● ●●●●●●●●●● ● ● ● ● ● ●●●●●●●●●●●●●● ● ● ● ● ● ●● ● ● 0 ● ● ●●●●●●●● ● ● ●

(x)) ● (x)) ● (x)) ● ● (x)) ● ● ● ● ● ● ● ● ● ● ● ● n n

● n ● ● ● ● ● n ● ● ● 20 ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● -20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ●● ● ●●●●●●●●●●● ● ●●●●●●●●●●●●● ● ● 0 ● ● ●●●●●●●●●● ● ● 0 ● ●●●●●●●●● ● ● ● ● ●●● ●● ● ● ● ● ●●● ● Im(ψ

● Im(ψ Im(ψ ● ● ● Im(ψ ● -50 ● -40 ● ● ● ● ● ● ● ● ● ● ● ● ● ● -20 -20 ● ● ● ● -60 ● ● ● ● -40 ● ● ● ● -40 ● ● ● -80 ● ● -100 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 -40 -20 0 20 40 60 -60 -40 -20 0 20 40

Re(ψn(x)) Re(ψn(x)) Re(ψn(x)) Re(ψn(x))

FIG. S3. (a)The real (Blue) and Imaginary (Orange) part of the right eigenvectors for the matrix B for the ﬁrst few eigenvalues. (b) Polar plots showing the real and imaginary parts of the eigenvectors for n ≥ 7. The polar plots are for n = 7, 9, 11.... The plots for even ordered eigenvectors (N = 8, 10...) are related to the eigenvectors of the previous eigenvectors by a reﬂection around x axis and hence are not plotted.