The intrinsic non-equilibrium nature of thermophoresis
TThe intrinsic non-equilibrium nature of thermophoresis
Shiling Liang, Daniel Maria Busiello, and Paolo De Los Rios
1, 2 Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland Institute of Bioengineering, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland
Exposing a solution to a temperature gradient can lead to the accumulation of particles on eitherthe cold or warm side. This phenomenon, known as thermophoresis, has been discovered more thana century ago and yet its microscopic origin is still debated. Here, we show that thermophoresis canbe observed in any system such that the transitions between different internal states are modulatedby temperature and such that different internal states have different transport properties. Weestablish thermophoresis as a genuine non-equilibrium effect, whereby a system of currents in realand internal space that is consistent with the thermodynamic necessity of transporting heat fromwarm to cold regions. Our approach also provides an expression for the Soret coefficient, whichdecides whether particles accumulate on the cold or on the warm side, that is associated to thecorrelation between the energies of the internal states and their transport properties, that insteadremain system specific quantities. Finally, we connect our results to previous approaches based onclose-to-equilibrium energetics. Our thermodynamically consistent approach thus encompasses andgeneralizes previous findings.
A solution in contact with a temperature gradient sup-ports the onset of a phenomenon known as thermophore-sis, thermodiffusion, or Ludwig-Soret effect [1, 2]. Thisis characterized by the net migration of particles towardseither the cold or warm side of the gradient, leading toa tilted stationary distribution [3]. The first evidence ofthermophoresis goes back to the work of Ludwig in 1856[4]. Since then, it has been observed in several differentsystems, such as colloidal suspensions [5], bimolecular so-lutions [6, 7], fluid mixtures [8] and DNA beads [9], justto cite some examples.Despite the overwhelming experimental evidence, acomprehensive microscopic theory of thermophoresis isstill lacking [10]. One of the main difficulties consists inthe fact that details about particle-solution interactionsseems to be non-negligible, being indeed necessary to pro-vide reliable predictions [11], and most of the theoreticalefforts rooted in system-dependent modeling have fallenshort of determining the essential ingredients responsiblefor the emergence of thermophoresis. Moreover, althoughthermophoresis feeds upon the imposed thermal gradient,the role of energy fluxes and their inevitable dissipationis a long-standing enigma [12].Heuristically, thermophoresis acts on the system as anexternal velocity drift, v . For diluted concentrations, itis usually assumed that v is proportional to the tem-perature gradient, ∂ x T [3]. This additional flux com-petes with standard diffusion, just as any other drift termwould, resulting in a non-uniform distribution at steady-state. Therefore, the total flux is: J = − D T c∂ x T − D∂ x c. (1)where c is the particle concentration, D and D T are, re-spectively, diffusion and thermodiffusion coefficients. Forthe sake of simplicity, we consider here a one-dimensionalsystem. The steady-state concentration, c ss , is usuallydetermined employing the zero-flux condition: ∂ x c ss c ss = − S T ∂ x T (2) with S T = D T /D is the so-called Soret coefficient. De-pending on the sign of S T , the particles accumulate onthe cold or warm side of the gradient.Eq. (2) relies on observations, and it is not obtainedfrom a microscopic theory. It is not known, for example,which are the system properties determining the sign ofthe Soret coefficient. Most importantly, it is still un-clear how to reconcile the zero-flux condition with thepresence of a thermal gradient and its thermodynamicconsequences ( e.g . heat transport, energy fluxes).Thermophoresis can be tackled in two different ways[3, 13]. Hydrodynamic arguments ascribe a dominantrole to the pressure difference caused by thermo-osmoticfluid flow around a particle [14]. On the contrary, ther-modynamic models are based on mesoscopic energeticanalyses, which account for the leading contributions tothermophoresis when particles are too small to experi-ence appreciable temperature differences on their sur-roundings.In this work we mostly focus on the thermodynamicapproach, building upon a preliminary observation pre-sented in [15], where thermophoresis emerged in a simplethree-state chemical system in the presence of a thermalgradient, as a consequence of the different diffusivitiesof internal states. Here we aim at formulating a generaltheory for particles with multiple internal states, pro-viding a microscopic derivation of the phenomenologicalequation, Eq. (1), finding an expression of the Soret coef-ficient as a function of the internal parameters. Further-more, we show that the onset of thermophoresis is inex-tricably related to the presence of non-vanishing fluxesin the system. Our approach extends to single-state par-ticles, highlighting that in this case the role of internalstates is played by different velocities in the complete un-derdamped description of the system. A previously re-ported result [16, 17], built on the assumption of a close-to-equilibrium regime, is also properly discussed withinour framework, by means of a thermodynamic energeticapproach. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Diffusivities, energies and thermophoresisDiscrete state-space
We first consider a system composed by particles withmultiple internal states in a solution, in the presence ofa thermal gradient, T ( x ). Despite being idealized, thismodel can be easily adapted to describe a large varietyof systems, ranging from polymer chains to enantiomers,and catalytic enzymes. In the case of polymer chains,for which thermophoresis has been observed [17], internalstates can be associated to different conformations. Anal-ogously, simple molecules can explore different isomers,each associated with an internal state, and enzymes canbe associated with the substrate, the product or alone.In short, multi-state particles capture a large array ofsystems with internal degrees of freedom.In what follows, we deal with linear reactions X i ↔ X j ,such as isomerization processes, even if it is straight-forward to generalize the model to multi-molecular re-actions ( e.g. catalysis ), as we will show later in thiswork. The probability of occupation of each internalstate i = 1 , . . . n (e.g. chemical species), P i , evolves ac-cording to Markovian dynamics and follows a reaction-diffusion equation [18]: ∂ t P i = D i ∂ x P i + n (cid:88) j =1 k ij P j , (3)where k ij is the rate at which state j transforms into state i , with the usual relation k ii = − (cid:80) j k ij to ensure prob-ability conservation, and D i is the diffusion coefficient ofstate i . In this formulation, temperature enters implic-itly in the kinetic rates, for which we do not provide yetan explicit expression. As a first approximation, we areconsidering diffusion coefficients that do not depend ontemperature, in order to show that thermophoresis canemerge even in this simple setting. Summing over theinternal states, we get rid of the information about thedynamics in the internal space obtaining: ∂ t P tot = (cid:88) i ∂ x ( D i P i ) = ∂ x ( ∂ x (cid:88) i D i P i (cid:124) (cid:123)(cid:122) (cid:125) − J tot ) . (4)where P tot = (cid:80) i P i . This equation describes the evolu-tion of the probability to be at x at time t , independentlyof the chemical states, but it does not provide a completesolution to the system, which requires instead as manyequations as the number of internal states. A simple example: two internal states
The onset of thermophoresis in this simple case hasbeen preliminary discussed in [15]. To fully character-ize the evolution of the system, we write the dynamical equations for P tot = P + P (analogous to Eq. (4)) andΠ = P − P : ∂ t P tot = ∂ x ( D P ) + ∂ x ( D P ) == D + D ∂ x P tot + D − D ∂ x Π (5) ∂ t Π = D − D ∂ x P tot + D + D ∂ x Π ++ ( k − k ) P tot − ( k + k ) Π . (6)Examination of Eq. (5) immediately reveals that a nec-essary condition for thermophoresis ( i.e. a non uniform P ss tot ) is that D (cid:54) = D , which we assume to be truethroughout this work. Solving Eq. (5) for Π, and intro-ducing it in Eq. (6), the exact stationary solution for thetotal probability, P ss tot , employing the zero-flux condition,is (see Appendix A1): ∂ x (cid:18)(cid:18) (cid:104) D (cid:105) eq P ss tot (cid:19) + (cid:18) D D k tot ∂ x P ss tot (cid:19)(cid:19) = 0 (7)where (cid:104) D (cid:105) = D k + D k k + k . If the transition rates are ofthe usual Arrhenius-like form: k k = e ( E − E ) /k B T ( x ) (8)where E i is the free energy of the state i , then (cid:104) D (cid:105) = (cid:104) D (cid:105) eq = 1 Z eq (cid:88) i D i e − E i /k B T ( x ) (9)with Z eq the equilibrium partition function.For slowly varying functions (small wavelength approx-imation, i.e. ∂ x P ss tot (cid:39) S T = ∂ T (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq (10)(see Appendix A1 for the detailed derivation of this ex-pression). The spatial and temperature dependence inthe effective diffusion coefficient, (cid:104) D (cid:105) eq , results from anaveraging procedure over internal states. Hence, (cid:104) D (cid:105) eq depends on x and T ( x ) through the kinetic rates k ij .Eqs. (5) and (6) represent a system of coupled diffusionequations (with a non-diagonal diffusion matrix) with anextra term in Eq. (6), linear in P tot and Π, that does notobey local continuity ( i.e. conservation) conditions andthat can thus be considered a source/sink. At equilib-rium, it vanishes because of detailed balance (it is indeednothing else than k P − k P ). In non-equilibrium con-ditions, instead, it does not vanish, indicating that thesystem is supporting non-equilibrium fluxes, that are de-ceptively hidden in the customary, zero-flux phenomeno-logical description, Eq. (2). In turn, this implies thatcurrents of the two species are present in the system.These fluxes are actually a thermodynamic necessityfor heat transport from warm to cold regions. The heatflux across the system is J E ( x ) = E ( − D ∂ x P ) + E ( − D ∂ x P ) (11) A B
FIG. 1: (A) Steady-state distribution when the covariance between energies and diffusion coefficients is positive. Particlesaccumulate on the cold side, denoting a positive Soret coefficients. The inset shows energies and diffusion coefficients of allstates. (B) For a negative covariance between energies and diffusion coefficients (shown in the Inset), particles accumulate onthe warm side, and the Soret coefficient is negative. The main panel shown the stationary profile. The temperature profile is k B T ( x ) = 0 . x + 1 in both cases. Using the relations between P , P , P tot and Π, it is pos-sible to show that, at steady-state and within the smallwavelength approximation for Eq. (7), the heat flux isdirected from the warm to the cold side: J E − ∂ x T = D D (cid:104) D (cid:105) eq E − E D − D ∂ T (cid:104) D (cid:105) eq P ss tot > Soret coefficient in the fast reaction limit
In the Appendix B1, we show that the expression forthe Soret coefficient derived above, Eq. (10), can be ob-tained for a general reaction network obeying Eq. (3), byemploying the fast reaction limit. In this approximation,which is equivalent to the small wavelength approxima-tion used for Eq. (7), the transition rates between chem-ical states, k ij , are much faster than diffusion, a realisticcondition in many experimental settings [19, 20], and thesystem locally relaxes to equilibrium. Eq. (10) can thenbe further developed, leading to (see Appendix B1 fordetails) S T = ∂ T (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq = (cid:104) ED (cid:105) eq − (cid:104) E (cid:105)(cid:104) D (cid:105) eq (cid:104) D (cid:105) eq T = Cov eq ( E, D ) (cid:104) D (cid:105) eq T (13)In Fig. 1, we show an illustrative example of a discrete-state system in which, inverting the covariance betweenenergies and diffusion coefficients, the steady-state dis-tribution P tot inverts its tilting accordingly.This results provides an insight into the physical ori-gin of the Soret coefficient and into its intimate structure. It highlights the intrinsic relation between thermophore-sis and how energies and diffusion coefficients are dis-tributed among the internal states. In particular, whenhigh energy states diffuse faster, particles tend to accu-mulate on the cold side, and viceversa. This observationmight stimulate a new avenue of research about the pos-sibility to design and control thermophoretic response ofbio-inspired chemical nanodevices [21, 22]. Continuous state-space
Discrete internal states are of course an approximationapplicable to systems with continuous internal variables (cid:126)q = { q , . . . q m } that are nonetheless localized most of thetime in a few regions because of, for example, deep min-ima of the potential energy function U ( (cid:126)q ). In this case,there are no discrete jumps from one internal state to theother. Instead, the system evolves in the internal spaceaccording to a diffusion equation, with diffusion constant∆( x ), and subject to a force − ∂ (cid:126)q U ( (cid:126)q ). The system alsoevolves in space according to a diffusion equation withdiffusion constant in space, D ( (cid:126)q ), that crucially dependson the internal state, as in the case of the discrete statesystem described before.The corresponding Fokker-Planck (in one spatial di-mension and with one internal degree of freedom for sim-plicity) is: ∂ t P = ∂ q ( P ∂ q U ( x, q ) + ∆( x ) ∂ q P ) (cid:124) (cid:123)(cid:122) (cid:125) − J q + ∂ x ( D ( q ) ∂ x P ) (cid:124) (cid:123)(cid:122) (cid:125) − J x (14)where the spatial and internal variable currents are high-lighted. Integrating over all internal degrees of freedom,the system can be described in terms of P tot ( x, t ) = (cid:82) dqP ( x, q, t ), which is the probability of finding a parti-cle in the position x at time t independently of the state q : ∂ t P tot = ∂ x (cid:18)(cid:90) dqD ( q ) P ( x, q, t ) (cid:19) = − ∂ x J tot ( x, q, t )(15)Eq. (15) does not provide a complete description of thesystem, since it results from the procedure of integratingout the information on q . As a consequence, the no-flux boundary condition, translating also in this case into J tot = 0 everywhere, deceptively hide the presence ofspatial fluxes for different values of the internal variable q . Once again, these fluxes are a consequence of the non-equilibrium setting and are necessary for heat transport.In Fig. 2 we show the stationary profile of P tot for thesimple case of a double-well potential, exhibiting a non-uniform distribution of particles in space, consistentlywith thermophoresis. FIG. 2: Stationary profile of the marginal distribution P tot ,showing an accumulation of particles on the cold side of thegradient (on the left, in this case), for three different valuesof diffusion coefficients. Upper inset - Full distribution inthe ( x, q ) space, with probability peaks corresponding to thelocation of wells in the subspace parametrized by q . Lowerinset - The double-well potential is sketched. Here, we set thediffusion coefficients to ∆( x ) = 1, and D ( q ) = α ( U ( q ) + 2 . α = 10 , − or + ∞ , as indicated by the legend. The limit of fast internal dynamics can be exploitedalso in this case, leading to (see Appendix B2): S T = Cov eq ( U, D ) (cid:104) D (cid:105) eq T with (cid:104)·(cid:105) eq = 1 Z q (cid:90) dq · e − U ( q )∆ (16)where Z q is the partition function of the internal space.The continuous case is thus equivalent to the discreteone. Underdamped picture for single-state particlesInternal states and phase-space
In the previous models, the diffusion constant was in-dependent from space to highlight that thermophoresisemerges by the interplay between currents in real and in-ternal space. Of course, if temperature depends on space, then, according to Einstein relation D ( x ) = k B T ( x ) /γ ( γ being the Stokes’ friction coefficient) also the diffusionconstant depends on x . Actually, this property aloneis enough to induce an accumulation of particles on thecold side (as if in the presence of a positive Soret coeffi-cient). Using a revised derivation of the diffusion equa-tion from the underdamped Kramers equation, we showthat also this effect is a consequence of the intrinsic non-equilibrium nature of a non-uniform temperature, withfluxes in the full phase-space.The Kramers equation for the evolution of the proba-bility P ( x, v, t ) is ∂ t P + v∂ x P = γm ∂ v (cid:18) vP + k B T ( x ) m ∂ v P (cid:19) (17)where m is the particle mass. It is possible to show (seeAppendix C1) that the correct parameter to perform aconsistent overdamped limit is the friction characteristictime-scale τ = m/γ . In particular, when τ − (cid:28)
1, therelaxation due to the friction is much faster than all othertime-scales in play, i.e. the system experiences a fasterequilibration in velocity space, and the system satisfiesthe following equation: ∂ t P = mγ ∂ x (cid:0) (cid:104) v (cid:105) eq P (cid:1) = ∂ x ( D ( x ) P ) (18)where P is the marginalized distribution obtainedby integrating P ( x, v, t ) over v , (cid:104)·(cid:105) eq = Z − (cid:82) dv · e − mv / (2 k B T ( x )) , that is the ensemble average over theequilibrium distribution in velocity space, with Z eq a nor-malization factor, and D ( x ) the overdamped diffusion co-efficient satisfying Einstein’s relation. Solving by usingthe zero-flux condition, as above, we obtain: S T = ∂ T (cid:0) (cid:104) v (cid:105) eq (cid:1) (cid:104) v (cid:105) eq = 1 T ( x ) (19)This is clearly an oversimplified model, whose aim is onlyto show that the Soret coefficient stems, again, from thepresence of internal states with different energies and dif-ferent transport coefficients. In this case, the internalstate variable is the velocity, and the internal state en-ergy is the kinetic energy. Clearly, higher kinetic energypositively correlate with faster transport resulting in apositive Soret coefficient. Non-equilibrium fluxes in phase-space
We integrated out variables associated with a fasterrelaxation to obtain equations for the total probabilityof finding a particle in position x at time t , as the ones inEq. (4) and Eq. (18). There are hidden non-equilibriumfluxes associated to these hidden degrees of freedom. Inthe previous simple case of two internal states, althoughthe total concentration was flux-less, there where spa-tial fluxes of the two states that, by conservation of theprobability, are accompanied by fluxes in the internalspace (Fig.3, upper panel). Analogously, when dealingwith the case of underdamped single-state particles, non-equilibrium fluxes take place in the whole phase-space, al-though they do not appear in the overdamped dynamicaldescription, Eq. (18). Their presence, however, is crucialboth to sustain a non-uniform stationary marginalizeddistribution P ( x ), which is a signature of the presence ofa thermophoretic effect, and to transport heat as dictatedby the laws of thermodynamics.The component of the current in position-space is J x ( x, v, t ) = vP ( x, v, t ) (see Appendix C2). Its integralover the velocity space is zero because of the zero-fluxcondition in space. However, J x ( x, v, t ) is not zero inthe whole phase-space ( x, v ), consistently with the non-equilibrium conditions. Indeed, we show in the AppendixC2, that the probability current of particles slower than | v | , for any | v | , J slow = (cid:90) | v |−| v | J x dv (cid:48) , (20)is always parallel to ∂ x T (thus directed from the coldto the warm side), implying that, because of the no-fluxcondition, the current accounting for particles faster than | v | , J fast = (cid:34)(cid:90) −| v |−∞ J x dv (cid:48) + (cid:90) + ∞| v | J x dv (cid:48) (cid:35) , (21)is always parallel to − ∂ x T (thus running from the warmto the cold side). Both of them vanish only for | v | = 0 and | v | → + ∞ , reaching their maximum absolute value at v ∗ = (cid:112) T ( x ) /m . Just as in the two-states case currentsclose in the internal space, here they close in velocityspace, with slow particles warming up on the warm sideand fast particles cooling down on the cold side (Fig.3,lower panel). The system thus picks up heat on the warmside, transports it across the system and releases it on thecold side. A detailed analysis of the total energy currentin position space shows that, to the leading order, atstationarity (see Appendix C3 for details) J Ex = (cid:90) ∞−∞ mv J x dv = − k B N ∂ x T (22)where N is the normalization factor of steady-state solu-tion. Soret coefficient and dimer formation
As an extension to the presented model, considerthe case of a non-diluted solution in which interactionsamong particles are not negligible. This picture allowsfor the formation of complex states, with a more complexenergy landscape. Here, we investigate the simple case
HOT COLDHOT COLDAB AB
Heat Heat ... ...
Heat Heat A B FIG. 3: (A) - Heat is absorbed on the warm side, hence driv-ing the system towards high-energy states. Afterwards, diffu-sion moves particles to the cold side, cooling them down andending up populating low energy states. Eventually, particlescome back to the warm side, hence restarting the cycle. (B)- As for panel (a), heat is absorbed on the warm side, andreleased on the cold side. When heated, particles overcomethe critical threshold v ∗ ( x ) and start moving towards the coldside. When cooled, they are driven below v ∗ ( x ), inverting thepreferential direction of the flux. In the full phase-space, thereare two identical cycles above and below the zero-velocity line,since energy depends on the absolute value of v . of dimer formation, as sketched in Fig. 4. The system isdescribed by the following reaction-diffusion equation: ∂ t c = 2 k − c − k + c + D ∂ x c ∂ t c = k + c − k − c + D ∂ x c (23)where c and c are, respectively, monomer and dimerconcentrations, satisfying the normalization condition c + 2 c = c tot , with c tot total concentration. The dis-sociation constant has the usual form K d ( x ) = k − ( x ) k + ( x ) ,where both association and dissociation rates depend onspace through temperature (as a reminder, the dissocia-tion constant has the dimensions of a concentration).Defining (cid:104)·(cid:105) eq = c tot (cid:80) n =1 · nc eq n , the Soret coefficientin the fast reaction limit is again of the form Eq. (13),and in particular (see Appendix D) S T = − F ( T, K d , c tot )1 − K d F ( T, K d , c tot ) ∂ T K d (24)where F ( T, K d , c tot ) = (cid:104) D (cid:105) − ( D − D ) c tot K − d g ( T, K d , c tot ), with g ( T, K d , c tot ) a posi-tive function. Since dimers are typically larger thanmonomers, hance D − D >
0, and the dissociationconstant increases with temperature (dimers tend todissociate at higher temperatures), the Soret coefficientcan be either negative or positive, with the overallconcentration of molecules higher on the warm or coldside, respectively. While an accumulation on the coldside intuitively follows the direction of the heat flow, itis also possible to conceive complex scenarios, wherebydimers are stabilized by contacts between unstructuredregions, as in proteins, hence increasing temperaturesmight stabilize the dimer state.In Fig. 4, we simulate the system of equation Eq. (D1),showing the appearance of thermophoresis. dimer monomer
ABC
FIG. 4: ((A) - Schematic reaction scheme of dimer forma-tion. (B) - Pictorial representation of dimer and monomerstationary populations in a thermal gradient. (C) - Steady-state profiles of dimer, monomer, and total concentration areshown. The system supports the onset of thermophoresis inthe presence of a thermal gradient.
Previously, we showed that single-state particles ex-hibit a positive Soret coefficient in dilute solutions(Eq. (19)). When particle-particle interactions becomenon-negligible, in non-dilute solutions, the potential for-mation of complex molecules may lead to an additionalcontribution to the Soret coefficient. The combination ofthese two effects might even result in an inversion of thethermophoretic response, as a function of particle con-centrations.
Discussion and conclusions
A theoretical understanding of thermophoresis has todate been elusive, despite the effect being well estab-lished. A source of confusion has for sure been thelack of a microscopic characterization of the Soret co-efficient, likely due to its apparent dependence on thesystem details. Furthermore, even though thermophore- sis is intrinsically a non-equilibrium effect, since it de-pends on the presence of a thermal gradient, its connec-tion with non-equilibrium statistical physics has not yetbeen fully established. As a matter of fact, several ap-proaches are based on a free-energy description, whichis formally inappropriate in a non-equilibrium scenario,and which can be recovered only from quasi-equilibriumor local-equilibrium assumptions, that must nonethelessbe justified on rigorous grounds.In this work we have tried to move a first step inthis direction, by firmly treating thermophoresis in theframework of stochastic thermodynamics, which is be-ing broadly accepted as the correct way to cast non-equilibrium phenomena. We could thus establish a few,important, facts about thermophoresis: • Thermophoresis emerges through the interplay be-tween transport in real space and temperature-modulated transitions in some internal space,which can be a chemical, conformational, or ve-locity space • The phenomenological approach to thermophore-sis, Eq.(2), is an approximation of the correct equa-tions, which is valid only in the fast reaction limit,corresponding to the local-equilibrium assumption • Eq. (2) is customarily solved with the no-flux con-dition, hiding the presence of currents for the un-derlying degrees of freedom, which are present bothin real and internal spaces • These currents are consistent with the non-equilibrium setting determined by the thermal gra-dient, and are actually a necessity for heat trans-port from warm to cold regions, as dictated by ther-modynamics • The Soret coefficient is related to the microscopicfeatures of the system through the correlation be-tween transport properties of each internal stateand its energyIn particular, we have provided here a general (albeitvalid only within the fast-reaction approximation) for-mula for the Soret coefficient, which proposes a bridgetoward its microscopic understanding, and rationalizesits dependence on a multitude of system-specific factors.For example, the diffusion coefficient of a given confor-mation (internal state) of the system might depend onits peculiar interactions with all the components of thesurrounding solvent, that can be derived only through acareful microscopic treatment. Nonetheless, Eq. 13 pro-vides the mathematical framework through which micro-scopic details must be assembled.In this respect, the connection to thermodynamic ap-proaches to thermophoresis deserve a special comment.Indeed, as shown in [16, 17], the thermodynamic deriva-tion of the Soret coefficient argues that S T = T − ∂ T G ,where G is the free-energy. However, in order to define afree-energy, each point in space should be in equilibriumwith a bath at the local temperature T ( x ). From stochas-tic thermodynamics arguments, dG = dQ − T dS , ( Q be-ing the heat and S the entropy) with dQ = 0 and dS = d ( − log P tot ), since thermophoresis is captured by a de-scription in terms of P tot , which follows a purely diffusiveequation, Eq. (4). Hence, S T = ∂ T S = − ∂ T P tot /P tot ,being indeed paired with a quantity encoding, throughthe derivative, information about spatial transport dueto thermal gradient. This mixed approach highlights thelink between our approach and previous ones.As already mentioned, our appraoch is not restrictedto the overdamped regime. As a matter of fact, through acareful and unambiguous derivation of the Smoluchowsky equation from the underdamped Kramers equation, wehave shown that also the presence of a diffusion constantthat depends on space, through its dependence on a non-uniform temperature field, goes hand-in-hand with thepresence of currents in phase space whose presence isnecessary for heat transport.We have also presented the extension of our model tothe case of non-dilute solutions with dimer formation,to highlight how our approach can be straightforwardlyextended to several other systems with internal states.Moreover, simple chemical systems could be experimen-tally tested, in order to verify and improve the theoreticalgrasp on the relationship between non-equilibrium fluxes,microscopic parameters, and thermophoresis. [1] M.A. Rahman and M.Z. Saghir. 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Physical ReviewResearch, 2(4):043257, 2020.[20] Avinash Vicholous Dass, Thomas Georgelin, FrancesWestall, Fr´ed´eric Foucher, Paolo De Los Rios,Daniel Maria Busiello, Shiling Liang, and Francesco Pi-azza. Equilibrium and non-equilibrium furanose selec-tion in the ribose isomerisation network. arXiv preprint arXiv:2010.13188, 2020.[21] J Calvin Giddings. Field-flow fractionation: analysisof macromolecular, colloidal, and particulate materials.Science, 260(5113):1456–1465, 1993. [22] PF Geelhoed, R Lindken, and J Westerweel. Ther-mophoretic separation in microfluidics. ChemicalEngineering Research and Design, 84(5):370–373, 2006. Appendix A: Simple example: two internal states1. The exact stationary solution
To find the exact solution of a two-state diffusion-reaction system, let us define two variables: the sum P tot = P + P and the difference Π = P − P . Using these two, we rewrite the time evolution equation: ∂ t P tot = D + D ∂ x P tot + D − D ∂ x Π ∂ t Π = D − D ∂ x P tot + D + D ∂ x Π+ ( k − k ) P tot − ( k + k )Π (A1)Moving in Fourier space, we introduce the Fourier transform of each function: F (Π( x, t )) = ˜Π( k, t ) F ( P tot ( x, t )) = ˜ P tot ( k, t ) (A2)hence, naming D = D + D and ∆ = D − D , we have the following set of equations:2 ∂ t ˜ P tot + D k ˜ P tot + ∆ k ˜Π = 0 (A3)2 ∂ t ˜Π + ∆ k ˜ P tot + D k ˜Π − F [( k − k ) P tot ] + 2 F [( k + k ) Π] = 0Solving the first equation, when k (cid:54) = 0, we have:˜Π( k, t ) = − ∂ t ˜ P tot ( k, t ) + D k ˜ P tot ( k, t )∆ k (A4)that, in real space, corresponds to:Π( x, t ) = − D ∆ P tot ( x, t ) + 1∆ ∂ t (cid:18)(cid:90) + ∞−∞ dξ P tot ( ξ )( x − ξ )sign( x − ξ ) (cid:19) (A5)A non-local term appears, which is the propagator of the one-dimensional Laplacian operator. It governs the diffusivedynamics of the system toward stationarity. However, the general solution of Π has to include an additional term:Π( x, t ) = − D ∆ P tot ( x, t ) + 1∆ ∂ t (cid:18)(cid:90) + ∞−∞ dξ P tot ( ξ )( x − ξ )sign( x − ξ ) (cid:19) + f ( x ) (A6)where f ( x ) is such that ∂ x f ( x ) = 0. Inserting (A6) into the second line of (A3), multiplying by ∆, and dividing by k sum = k + k , we obtain: − D + D k sum ∂ t P tot + (cid:18) D k k sum + D k k sum (cid:19) P tot + ∂ t (cid:18) (cid:90) dξ P tot ( ξ, t )( x − ξ )sign( x − ξ ) (cid:19) ++ 1 k sum ∂ t (cid:18) (cid:90) dξ P tot ( ξ, t )( x − ξ )sign( x − ξ ) (cid:19) + D D k sum ∂ x P tot + ( D − D ) f ( x ) = 0 (A7)This equation is directly in real space. We remark that we have divided by k sum , in order to isolate the term f ( x ).Since we also know that the second spatial derivative of this function has to vanish, we can derive twice with respectto x : ∂ t P tot − ∂ x (cid:18) (cid:18) D k k sum + D k k sum (cid:19) P tot (cid:19) ++ ∂ x (cid:18) k sum D D ∂ x P tot (cid:19) ++ ( D + D ) ∂ x (cid:18) k sum ∂ t P tot (cid:19) + ∂ x (cid:18) k sum ∂ t (cid:90) dξ P tot ( ξ, t )( x − ξ )sign( x − ξ ) (cid:19) = 0 (A8)This form constitutes the exact solution, without any approximation. The first line is the same equation that appearsin the fast reaction limit, derived in detail in the next Section. The second line is the correction to compute thesteady state, P ss tot , when the chemical reactions are not faster than diffusion. The last line controls the evolution of P tot towards the steady state, and it contains the kernel of long-range interactions. At this point, we note that thefast reaction limit is, in general, a good approximation, since the only correction, at stationarity, is a fourth-orderspatial derivative of P ss tot . In Fig. 5 we compare the steady state profile in the fast reaction limit with two choices ofparameters, D = 10 − (reactions faster than diffusion), and D = 10 − (reactions slower than diffusion).For slowly varying functions, i.e. in the small wavelength approximation, corresponding to ∂ x P ss tot (cid:39)
0, (A8) takesthe following form at stationarity: ∂ x (cid:18) (cid:18) D k k sum + D k k sum (cid:19) P ss tot (cid:19) = 0 (A9)If the transition rates are of the usual Arrhenius-like form: k k = e ( E − E ) /k B T ( x ) (A10)where E i is the free energy of the state i , we define the equilibrium ensemble average of the diffusion coefficient: (cid:104) D (cid:105) eq = 1 Z eq (cid:88) i D i e − E i /k B T ( x ) (A11)with Z eq the equilibrium partition function.Then, employing the zero-flux condition, we have the following equation for the stationary state: P ss tot ∂ x (cid:104) D (cid:105) eq + (cid:104) D (cid:105) eq ∂ x P ss tot = 0 (A12)which leads to the following identification of the Soret coefficient: ∂ x P ss tot P ss tot = − ∂ T (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq ∂ x T (A13)
2. The energy flux from the warm side to the cold side
For two-state particles diffusing in a temperature gradient, the energy flux associated to transport of heat acrossthe system is: J E = − E D ∂ x P − E D ∂ x P (A14)where − D i ∂ x P i is the diffusive probability flux for the state i . Writing P and P as functions of P tot and Π, we have: J E = − (cid:18) E D + E D (cid:19) ∂ x P tot − (cid:18) E D − E D (cid:19) ∂ x Π (A15)Reminding that, at steady-state, Π ss = − P ss tot D / ∆ (see the previous subsection), after some algebra, we have: J E = 1∆ ( E − E ) D D ∂ x P ss tot (A16)Spelling out ∂ x P ss tot by inverting (A13), we finally have: J E − ∂ x T = E − E D − D D D ∂ T (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq P ss tot = D D E − E D − D S T P ss tot (A17)0 fast reaction limit s l o w r eac ti on s f a s t r eac ti on s FIG. 5: The steady state profile of total probability P ss tot = P ss A + P ss B is compared with two different choices of parameters: D = 10 − , for which reactions are slower than diffusion, and D = 10 − , for which reactions are faster than diffusion. Thetemperature gradient is set to T ( x ) = 0 . x ), for x ∈ [0 , Appendix B: Soret coefficient in the fast-reaction limit1. Discrete chemical space
Let us start from the reaction-diffusion system defined in the main text: ∂ t P i = D i ∂ x P i + n (cid:88) j =1 k ij P j , (B1)Let us suppose that chemical reaction are faster than diffusion. Starting from Eq. (B1), we can employ a standardtime-scale separation analysis. First, we explicit the time scale of reaction rate, k ij = ˜ k ij /(cid:15) , with (cid:15) (cid:28)
1. Second, wepropose a solution in the following form: P i ( x, t ) = P (0) i ( x, t ) + (cid:15)P (1) i ( x, t ) + O ( (cid:15) ) (B2)Plugging this expression into Eq. (B1), and solving order by order, we find the following zeroth order equation: n (cid:88) j =1 ˜ k ij P (0) j ( x ) = 0 (B3)This means that P (0) i ( x, t ) can be written as: P (0) i ( x ) = 1 Z e − EikBT ( x ) π ( x, t ) = P (eq) i ( x ) π ( x, t ) (B4)where T ( x ) is the local temperature at position x , Z = (cid:80) i e − E i /k B T ( x ) is the partition function. In other words, it isthe product between Boltzmann distribution in chemical space and a generic time-dependent function.At first order, summing over all chemical states, the system satisfy: ∂ t π ( x, t ) = n (cid:88) j =1 D j ∂ x (cid:16) P (eq) j ( x ) π ( x, t ) (cid:17) (B5)1Noting that π ( x, t ) = (cid:80) j P (0) j ( x ) = P tot ( x ), up to the leading order in (cid:15) , from Eq. (B5), we obtain: S T = ∂ T (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq = 1 (cid:104) D (cid:105) eq ∂ T (cid:18) (cid:80) i D i e − E i /k B T Z (cid:19) = 1 (cid:104) D (cid:105) eq Z k B T (cid:32) Z (cid:88) i D i E i e − E i /k B T − (cid:88) i D i e − E i /k B T (cid:88) i D i e − E i /k B T (cid:33) = (cid:104) ED (cid:105) eq − (cid:104) E (cid:105) eq (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq k B T = Cov eq ( E, D ) (cid:104) D (cid:105) eq k B T (B6)with the equilibrium ensemble average defined as: (cid:104) D (cid:105) eq = 1 Z (cid:88) i D i e − EikBT ( x ) (B7)Hence, the Soret coefficient is the ensemble covariance of energies and diffusion coefficients. This provides somephysical insights into the thermophoresis problem from an energetic perspective. If energy and diffusion coefficientsare positively correlated, i.e. particle in high-energy states diffuse faster, then the Soret coefficient is positive, and theparticles tend to move from the hot region to the cold region. The opposite motion happens when there is a negativecorrelation.
2. Continuous chemical space
A system may experience a continuum of possible internal states, and the reaction-diffusion equation results as acoarse-grained description of transition among local minima. When this approximation is not valid, the completedynamics of the system can be captured by a Fokker-Planck equation, as introduced in the main text: ∂ t P = ∇ q (cid:18) γ q P ∇ q U q + ∇ q ( D q P ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) − J q + ∂ x ( ∂ x ( D x P )) (cid:124) (cid:123)(cid:122) (cid:125) − J x . (B8)where P ≡ P ( x, q ), is the probability distribution in the position-chemical space. We aim at investigating the fastreaction limit in this context, i.e. the flux acting on chemical space, J q → ˜ J q /(cid:15) , with (cid:15) (cid:28)
1, is much stronger thanthe one acting on real space, J x . This condition is also equivalent to the following assumption: D x D q = (cid:15) (cid:28) γ q D q = k B T ( x ) (B9)where the last equality corresponds to the Einstein relation. Guessing a solution of the following form: P = P (0) + (cid:15)P (1) + O ( (cid:15) ) . (B10)at the zeroth order, we have: 0 = − ∂ q P (0) − k B T ( x ) ∂ q ( P (0) ∂ q U q ) (B11)By solving this equation, we have: P (0) ( x, q ) = P ( x ) exp (cid:18) − U q k B T ( x ) (cid:19) . (B12)2where P ( x ) has to be determined solving the first order equation, and the other factor is the equilibrium distributionin chemical space. At first order, integrating over the chemical space by employing the zero-flux condition, we have: ∂ x P ( x ) Z q (cid:124) (cid:123)(cid:122) (cid:125) Φ( x ) Z q (cid:90) ∞−∞ d q P ( x ) a ( q ) exp (cid:18) − U q k B T (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) D eff / ˜ D x = 0 , (B13)where Φ( x ) = (cid:82) d d q P ( x, q ) is the marginal distribution along x , while the other term is the average of the D x overchemical space. Writing the ensemble average in chemical space as (cid:104)·(cid:105) q = Z q (cid:82) q d q · exp (cid:16) − U q k B T (cid:17) , we obtain ∂ x (Φ( x ) (cid:104) D x (cid:105) q ) = 0 , (B14)which is the same as the discrete-state case. The Soret coefficient can be equally written in covariance form: S T = ∂ T (cid:104) D x (cid:105) q (cid:104) D x (cid:105) q = Cov eq ( E, D x ) (cid:104) D x (cid:105) eq k B T . (B15) Appendix C: Smoluchowski equation from Kramers equation1. Kramers equation and time-scales
Let us start from the Kramers equation: ∂ t P ( x, v, t ) + v∂ x P ( x, v, t ) = γm ∂ v (cid:18) vP ( x, v, t ) + T ( x ) m ∂ v P ( x, v, t ) (cid:19) (C1)We assume that the system is operating in the overdamped regime, i.e. in a region of parameters where the time-scaleassociated to the friction, τ − = γ/m is much faster than all the others in play. It is then natural to employ atime-scale separation procedure to develop an equation describing the dynamics at the leading order.In order to understand the correct scaling of (C1) with respect to the expansion parameter τ (cid:28)
1, we perform thefollowing change of variables: x = x √ γ v = v √ m (C2)Then, in terms of these new variables, (C1) becomes: τ ∂ t P (x , v , t ) + √ τ v ∂ x P (x , v , t ) = ∂ v (cid:18) v P (x , v , t ) + T (x) ∂ v P (x , v , t ) (cid:19) (C3)In the limit of small 1 /τ , the probability distribution can be written as: P (x , v , t ) = P (x , v , t ) + √ τ P (x , v , t ) + τ P (x , v , t ) + O (cid:16) τ / (cid:17) (C4)since the smallest order appearing in the modified Kramers equation (C3) is proportional to √ τ , which is then thenatural expansion parameter to be adopted. a. Zeroth order Substituting (C4) in (C3) we obtain a set of equations at different orders in √ τ . The zeroth order equation is equalto: ∂ v (cid:18) v P (x , v , t ) + T (x) ∂ v P (x , v , t ) (cid:19) (C5)It is easy to verify that the solution is in the form P (x , v , t ) = e − v22 T (x) Φ (x , t ).3 b. First order The equation resulting from the first order terms is:v ∂ x P (x , v , t ) = ∂ v (cid:18) v P (x , v , t ) + T (x) ∂ v P (x , v , t ) (cid:19) (C6)Introducing the expression for P , and guessing the P has a similar form, i.e. P = e − v22 T (x) Φ (x , v , t ), we get: e − v22 T Φ v T ∂ x T + v e − v22 T ∂ x Φ = ∂ v (cid:18) v P (x , v , t ) + T ∂ v P (x , v , t ) (cid:19) = ∂ v (cid:18) T e − v22 T ∂ v Φ (cid:19) (C7)where, for sake of clarity, we have not reported all the dependence on x and v.Guessing a solution of the form ∂ v Φ (x , v , t ) = − ∂ x Φ (x , t ) + F (x , v , t ), we obtain: e − v22 T Φ v T ∂ x T = ∂ v (cid:18) T e − v22 T F (cid:19) (C8)Integrating this equation between −∞ and v (cid:48) , we get: T e − v (cid:48) T F = − e − v (cid:48) T Φ (cid:0) v (cid:48) + 2 T (cid:1) T ∂ x T → F = − v (cid:48) + 2 T T Φ ∂ x T (C9)where we assumed that the probability distribution has no divergent behavior at any order. Then, we have thefollowing solution for the full probability distribution: P (x , v , t ) = e − v22 T (cid:26)(cid:0) Φ + √ τ f (x , t ) (cid:1) − √ τ (cid:20) v ∂ x Φ + (cid:18) v T + v T (cid:19) Φ ∂ x T (cid:21)(cid:27) + O ( τ ) (C10)where f (x , t ) is an arbitrary function acting as a perturbation of Φ (x , t ).Moreover, P (x , v , t ) is the full probability distribution of x and v. However, we are interested in deriving an effectiveequation for the evolution of the the marginal distribution of x and t . In fact, the time-scale expansion we haveperformed is compatible with the situation in which the variables v thermalize faster than x. Using the standard ideaof the time-scale separation, we integrate over v, in order to understand what is the effective probability distributionwe would like to describe, P (x , t ), in terms of the full state space: P (x , t ) = (cid:90) + ∞−∞ d v P (x , v , t ) = (cid:90) + ∞−∞ d v e − v22 T (x) Φ (x , t ) = Φ (x , t ) (cid:104) (cid:105) (C11)where, from now on, we adopt the following notation: (cid:104)·(cid:105) = (cid:82) + ∞−∞ · e − v22 T (x) d v. The integration can be easily performedby noting the parity of the function involved. c. Second order Going up to the second order in √ τ , and using the general expressions for P and P , we obtain the followingequation: ∂ t (cid:16) e − v22 T Φ (cid:17) + v ∂ x (cid:16) e − v22 T Φ (cid:17) = ∂ v (cid:18) v P + T ∂ v P (cid:19) (C12)Since we want to find the dynamical evolution of the function P (x , t ) = Φ (x , t ) (cid:104) (cid:105) defined in (C11), we integrate(C12) over the domain of v, obtaining: (cid:104) (cid:105) ∂ t Φ (x , t ) + ∂ x ( (cid:104) vΦ (x , v , t ) (cid:105) ) = ∂ t P (x , t ) + ∂ x ( (cid:104) vΦ (x , v , t ) (cid:105) ) = 0 (C13)where we have used again the non-singularity condition of the probability distribution.Here, we evaluate the following integral: (cid:104) vΦ (x , v , t ) (cid:105) = −(cid:104) v (cid:105) ∂ x Φ − (cid:104) v (cid:105) Φ T (x) ∂ x T (x) − (cid:104) v (cid:105) Φ T (x) ∂ x T (x) (C14)4Notice that the function f introduced in (C10) disappears after the integration over v both in the evaluation of P ,(C11), and in the equation above. Then, f has no effect in the limit of thermalizing velocities (i.e. overdamped) weare considering.Using the fact that (cid:104) v (cid:105) and (cid:104) v (cid:105) can be evaluated explicitly, and that the following relation holds: (cid:104) v (cid:105) = 3 (cid:104) v (cid:105) T (x),we get: (cid:104) vΦ (x , v , t ) (cid:105) = −(cid:104) v (cid:105) ∂ x Φ − (cid:104) v (cid:105) T (x) ∂ x T (x) (C15)Moreover, we highlight that: ∂ x (cid:104) v (cid:105) = (cid:104) v (cid:105) ∂ x T (x)2 T (x) = (cid:104) v (cid:105) ∂ x T (x) T (x) (C16)Substituting this equality in (C15), we finally get: (cid:104) vΦ (x , v , t ) (cid:105) = −(cid:104) v (cid:105) ∂ x Φ (x , t ) − Φ (x , t ) ∂ x (cid:104) v (cid:105) = − ∂ x (cid:0) (cid:104) v (cid:105) Φ (x , t ) (cid:1) (C17)Then, (C18), in terms of the marginal probability distribution P (x , t ), becomes: ∂ t P (x , t ) = ∂ x (cid:20) ∂ x (cid:18) (cid:104) v (cid:105)(cid:104) (cid:105) P (x , t ) (cid:19)(cid:21) = ∂ x (cid:2) ∂ x (cid:0) (cid:104) v (cid:105) eq P (x , t ) (cid:1)(cid:3) (C18)which is the standard Smoluchowski equation, where D (x) = (cid:104) v (cid:105) eq . Note that we identified (cid:104) v (cid:105) / (cid:104) (cid:105) = (cid:104) v (cid:105) eq , thatis the equilibrium ensemble average of v . d. Conclusions Hence, the origin of the thermophoretic effect for hard spheres, with no internal states, diffusing in a temperaturegradient has to be found in the fact that spheres with different velocities have different transport properties. In otherwords, the ensemble of possible states in the velocity space plays the same role of the internal states of a molecule. Itis important to note that the integration over the velocity states is crucial, since it allows us to associate to the sameposition x a plethora of states with different velocities: this is why these latter assume the same flavour of internal(e.g. configurational, energetic) states.By writing down the expression of D (x), and mapping back the final equation to the original variables ( x, v ), weobtain the following standard Smoluchowski equation with a diffusion coefficient satisfying Einstein relation, as forthe original Kramers equation we started with: ∂ t P ( x, t ) = mγ ∂ x (cid:0) (cid:104) v (cid:105) eq P ( x, t ) (cid:1) = ∂ x ( D ( x ) P ( x, t )) (C19)where D ( x ) = k B T ( x ) /γ is the standard overdamped diffusion coefficient. e. Beyond the second order - determination of f ( x, t ) The second order equation (C12) can be solved by looking for a solution of the form P = e − v22 T x) Φ (x , v , t ), andintegrating over v (cid:48) between −∞ and v, as before. This leads to: (cid:104) (cid:105) (v) ∂ t Φ + ∂ x (cid:104) vΦ (cid:105) (v) = T e − v22 T ∂ v Φ (C20)employing the non-singular behaviour of P , where (cid:104)·(cid:105) (v) = (cid:82) v −∞ . Expressing (cid:104)·(cid:105) (v) in terms of (cid:104)·(cid:105) , we obtain: (cid:104) (cid:105) (v) ∂ t Φ + ∂ x (cid:104) vΦ (cid:105) (v) = 12 (cid:18) (cid:18) v √ (cid:19)(cid:19) (cid:18) (cid:104) (cid:105) ∂ t Φ + ∂ x (cid:104) vΦ (cid:105) (cid:19) + e − v T I (x , v) (C21)where I (x , v) is the remaining integral that does not contain exponential terms. Using the Smoluchowski equation,(C18): ∂ v Φ = 1 T I (x , v) (C22)5Writing the third order equation, we get: ∂ t P + v ∂ x P = ∂ v (v P + ∂ v P ) (C23)Integrating over v in the whole range of the velocities (from −∞ to + ∞ ), we obtain: ∂ t f = − (cid:104) (cid:105) ∂ x (cid:18) T (x) (cid:104) ∂ v Φ (cid:105) (cid:19) = − (cid:104) (cid:105) ∂ x (cid:104) I (x , v) (cid:105) (C24)Spelling out the integral, we have: ∂ t f = T ∂ f + 3 ∂ x T ∂ x f + 3 f ( ∂ x T ) + 2 T ∂ T T (C25)If at t = 0, f (x ,
0) = 0, imposing an initial condition only on Φ satisfying the Smoluchowski equation, then f = 0 atall times and x, consistently with the equation above.
2. Fluxes and velocity fronts
Let us study probability fluxes. The Kramers equation is a two-dimensional Fokker-Planck equation whose flux isa two-dimensional vector: ∂ t P (x , v , t ) = (cid:0) ∂ x ∂ v (cid:1) (cid:18) J x J v (cid:19) = (cid:0) ∂ x ∂ v (cid:1) (cid:32) √ γ J x √ m J v (cid:33) (C26)writing the gradient in real space ( x, v ). Here, we have: (cid:18) J x J v (cid:19) = (cid:32) − vP ( x, v, t ) γm (cid:16) vP ( x, v, t ) + k B T ( x ) m ∂ v P ( x, v, t ) (cid:17) (cid:33) = (cid:126)J (C27)Since velocity-space thermalizes much faster by assumption, we are interested in the position of particles. Thus, wewill focus our attention only at the first component, J x .If we integrate J x over the full velocity-space, by construction, we recover the standard flux of the Smoluchowskiequation, J S . Here, the second component J v does not play a role since the limit of equilibrated velocities is implicitlyconsidered. J S = mγ ∂ x (cid:18) (cid:104) v (cid:105)(cid:104) (cid:105) P (cid:19) = (cid:104) vΦ (cid:105) = (cid:90) + ∞−∞ dvJ x (C28)where the first equality has been derived above for scaled variables, and the last one has been obtained using theparity of the functions involved. Here, we are considering the probability distribution up to the first order in √ τ .If we include the zero-flux boundary condition, at the stationary state, we have J S = 0, which is exactly the samecondition we used to derive the Soret coefficient. Remembering that fast and slow particles have different transportproperties, leading to the appearance of thermophoresis, for a system in the stationary state, we have: J slow = (cid:90) | v (cid:48) |−| v (cid:48) | dvJ x J fast = (cid:90) −| v (cid:48) |−∞ dvJ x + (cid:90) + ∞| v (cid:48) | dvJ x J fast = − J slow (C29)since J fast + J slow = J S = 0. Then, hot and cold fluxes balance each other for any value of | v (cid:48) | , in order to have a zeroflux at the level of Smoluchowski equation. However, the system still preserves the presence of microscopic fluxes inthe velocity space.By developing the integral, using (C11), and remembering that S T = 1 /T in this case, J slow has the followingexpression: J slow = ∂ x T (cid:112) πT /m | v (cid:48) | e − m | v (cid:48)| kBT P ∝ ∂ x T (C30)Hence, J slow is parallel to ∂ x T , while J fast is parallel to − ∂ x T . Furthermore, J slow is zero only at | v (cid:48) | → | v (cid:48) | → ∞ , reaching a maximum at the critical velocity front | v (cid:48) | = v ∗ ( x ) = (cid:112) T ( x ) /m .6
3. Energy flux
The kinetic energy flux accounts for the amount of energy transported across the system. To show that our modelis thermodynamically consistent, we show that: J Ex = (cid:90) + ∞−∞ dv mv J x = − D ( x )2 P ∂ x T ∝ − ∂ x T (C31)This result has been obtained after some algebra, using (C11), and the fact that S T = 1 /T . The minus sign indicatesthat the energy flows from the warm to the cold side, as dictated by thermodynamics. At stationarity, P assumes itstationary value, and J Ex becomes as the formula shown in the main text. Appendix D: Soret coefficient and dimer formation
The system is described by the following reaction-diffusion equation (see main text): ∂ t c = 2 k − c − k + c + D ∂ x c ∂ t c = k + c − k − c + D ∂ x c (D1)where c and c are, respectively, monomer and dimer concentrations, satisfying the normalization condition c +2 c = c tot , with c tot total concentration. The dissociation constant has the usual form K d ( x ) = k − ( x ) k + ( x ) , where both associationand dissociation rates depend on space through temperature.Performing the fast reaction limit, the Soret coefficient is defined by: S T ∂ x T = ∂ x (cid:104) D (cid:105) eq (cid:104) D (cid:105) eq = − ∂ x c tot c tot (D2)where (cid:104)·(cid:105) = c − tot (cid:80) n =1 · nc eq n , where n is the stoichiometric number, which is 1 for monomers, and 2 for dimers. Here, c eq n satisfies the chemical part of (D1), resulting in the following expression: c eq1 = c tot
21 + (cid:112) c tot /K d ) c eq2 = c tot − c eq1 ) (D3)We remind that c tot and K d depend on space, thus also c eq1 and c eq2 do. Let us evaluate the expression of ∂ x (cid:104) D (cid:105) eq : ∂ x (cid:104) D (cid:105) eq = ∂ x (cid:18) D c eq1 + 2 D c eq2 c tot (cid:19) = ( D − D ) ∂ x (cid:18) c eq1 c tot (cid:19) = ( D − D ) ( c eq1 /c tot ) (cid:112) c tot /K d ) ∂ x (cid:18) c tot K d (cid:19) (D4)By inverting the last equality of (D2), we have: ∂ x c tot = c tot ( D − D ) ( c eq1 /c tot ) (cid:112) c tot /K d ) ∂ x (cid:18) c tot K d (cid:19) (cid:104) D (cid:105) eq = D − D (cid:104) D (cid:105) eq g ( T, K d , c tot ) ∂ x (cid:18) c tot K d (cid:19) (D5)where g is a positive function. Naming g ( D − D ) / (cid:104) D (cid:105) eq ≡ F , and recalling that D > D in this case, we have thatalso F is a positive function. Hence, solving (D5) for ∂ x c tot , we have: ∂ x c tot = − F c tot − K d F ∂ x K d (D6)Putting this expression back into (D2), we have the expression for the Soret coefficient shown in the main text: S T = F − K d F ∂ T K dd