Stationary currents in long-range interacting magnetic systems

We construct a solution for the $1d$ integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in [G. B. Giacomin, J. L. Lebowitz,"Phase segregation dynamics in particle system with long range interactions", Journal of Statistical Physics 87(1) (1997)]. This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials. The microscopic system is in contact with reservoirs of fixed magnetization and infinite volume, so that their density is not affected by any exchange with the bulk in the original Kawasaki dynamics. At the mesoscopic level, this condition is mimicked by the adoption Dirichlet boundary conditions. We derive the stationary equation of the model starting from the Lebowitz-Penrose free energy functional defined on the interval $[-\varepsilon^{-1},\varepsilon^{-1}]$, $\varepsilon>0$. For $\varepsilon$ small, we prove that below the critical temperature there exists a solution that carries positive current provided boundary values are opposite in sign and lie in the metastable region. Such profile is no longer monotone, connecting the two phases through an antisymmetric interface localized around the origin. This represents an analytic proof of the existence of diffusion along the concentration gradient in one-component systems undergoing a phase transition, a phenomenon generally known as uphill diffusion. However uniqueness is lacking, and we have a clue that the stationary solution obtained is not unique, as suggested by numerical simulations.


INTRODUCTION
The aim of this paper is to study Fick's law of transport in one-component systems undergoing a second order phase transition. In this context, it represents a step forward towards the establishment of a well posed theory for diffusion along the gradient (uphill diffusion). Fick's law relates the flux J of a given substance to the gradient of its concentration ρ, which we suppose to be a differentiable function of the position in [0, L]: Nernst [2], Onsager [3] and especially Darken [4][5][6], who performed an acknowledged experiment in the late 40's. His setup consisted of pairs of doped steels (Fe-Si with a different wt. % of silicon, Fe-Si and Fe-Mn or Fe-Si and Fe-Mo) containing a small difference in the carbon concentration at the edges. The steels were welded together and eventually held in a furnace in order to let diffusion occur. In fact, it was observed that carbon diffused following the gradient in the mixtures with slightly differences in carbon concentration. This is shown in Figure 1, which refers to the Fe-Si-Mn compound after two weeks the experiment started. This counterintuitive behavior has a microscopic origin: in fact, silicon decreases the chemical affinity of carbon, while manganese increases it. This results in a driving force that acts in the opposite direction with respect to the concentration gradient and it might beat the gradient, if the difference in the carbon concentration at the edges is small. Such mechanism, which actually sustains uphill diffusion, works until dopants penetrate the weld; then, "standard" diffusion regime is restored. In formulae, this can be modeled by replacing (1.1) with a vectorial relation: where n > 2 is the number of components of the system, i refers to a given component, c i is the molar concentration of i, L ij represents the Fick diffusivity of i given the presence of j, while the µ i 's are chemical potentials. (1.2) describes a system of n − 1 linearly independent equations, because of the Gibbs-Duhem relation for chemical potentials [7,8].
Very surprisingly, numerical simulations suggest that uphill diffusion may also occur also in one-component systems undergoing a phase separation. Colangeli et al. considered in [9] a 1d stochastic automaton describing a dissipative system of particles interacting at large distances. After a transient, a stationary state with non zero current emerges and, moreover, a region in which diffusion follows the concentration gradient can be spotted tuning the characteristic parameters of the system. Similar results have been obtained by running a Kawasaki dynamics for an Ising spin chain with Kac potentials below the critical temperature, in which particles located at the edges may flip according to assigned rates, in order to mimic interactions with reservoirs of infinite volume and opposite magnetization [10]. When the magnetizations of the reservoirs are suitably chosen, the flux follows the "magnetization gradient". The resulting steady profile, called bump, is no longer monotone and connects the two boundary values through an interface that is localized in the nearby of one of the edges, randomly selected by dynamics. Colangeli et al. [11] obtained analogue numerical results for the 2d nearest neighbors Ising model. The microscopic mechanism underlying uphill diffusion in one-component systems has not a chemical origin. We speculate that the "force" that counteracts the gradient is provided in this case by the separation of phases; however, we believe that such state is in fact metastable, meaning that bumps are local minima for the corresponding Gibbs free energy, but not global ones. Hence, after a transient, the flux should reverse to be directed from the state at higher magnetization to the state at lower magnetization. Nevertheless, such inversion does not take place in the time considered for simulations.
Here we prove analytically the occurrence of uphill diffusion considering the model that is the continuous limit of the Ising chain with Kac potentials and Dirichlet boundary conditions. Our starting point is the Lebowitz-Penrose free energy functional, that is a non-local version of the scalar Ginzburg-Landau functional and that we postulate to describe the Physics of the system at the mesoscopic level. This represents the intermediate scale between the microscopic, discrete chain and the macroscopic model, which is obtained letting the size of the system diverge. We know that in this case the phase diagram (i.e. the free energy density vs. magnetization diagram) has a global minimum for β < 1, β the inverse temperature in our units, while for β > 1 the graph is flat in [−m β , m β ], m β the positive solution of the mean field equation m = tanh (βm). This indicates the occurrence of a phase transition. Any value in (−m β , m β ), the so called spinodal region, is then forbidden, so that any stationary profile containing values smaller of −m β and larger than m β must be discontinuous. At the mesoscopic level, the spinodal region is actually available and the discontinuity is replaced by a smooth interface, as proved by De Masi et al. for the free-boundary Stefan problem [14][15][16][17]. However, the discontinuity is recovered when the hydrodynamic limit is performed. We then fix β > 1 (β = 1 the critical inverse temperature in the mean-field model) and look for stationary solutions ofṁ = − ∂ ∂x I in the space of bounded antisymmetric functions, I being the local current which is supposed to be proportional to the functional derivative of the free energy. Thus, we reduce to the problem ∂ ∂x I = 0 in the finite interval [−ε −1 , ε −1 ], ε > 0 fixed, which turns out to be an integro-differential equation [1]. The corresponding Dirichlet problem has been already studied in [10] and [13], although in the presence of an external, antisymmetric magnetic field. In that case it has been proved that, whatever the intensity of this field, the provided "external force" cannot reverse the flux when the positive boundary condition is in the interval (m * (β) , m β ), m * (β) = 1 − 1/β the positive saddle point of the mean field free energy ( Figure 1). Figure 3. The mean field free energy at β > 1.
We solve the stationary equation and prove that our resulting profile actually carries positive current. For x ≥ 0 the solution firstly increases, jumping "instantly" from zero to m β , then decreases to the metastable value at the right boundary; however, the region in which the current flows in the "wrong" direction reduces to a set of zero Lebesgue measure in the hydrodinamic limit, so that Fick's Law actually holds almost everywhere (w.r.t. the Lebesgue measure). The weak spot of the analysis is that our solution is supposedly unstable; in fact, numerical simulations suggest that bumps should be stable points for the gradient dynamics.
The stationary Stefan problem in bounded domains has been already considered by De Masi, Presutti and Tsagkarogiannis in [22], despite Neumann conditions have been adopted there. Apart from technicalities in the proof, the two approaches are quite different, since in the Neumann setting the magnetization profile naturally selects the boundary values imposed by the choice of current. However, as clarified in that work, there are solutions of the mesoscopic Neumann version of problem that converge to any solution of the Dirichlet problem as ε → 0.
, m Λ being the magnetization density of the bulk and m Λ c the magnetization of the reservoirs. Our starting point is the mesoscopic Lebowitz-Penrose free energy functional at zero external magnetic field, that is and S (m) is the standard binary entropy for an Ising spin system: J is a probability kernel that actually depends on the distance between two points. The assumptions made onJ are precisely listed below: We treat expressions (2.1), (2.2) as primitive quantities, by postulating them to define our model at a mesoscopic level. Indeed, it can be shown that this is precisely what one obtains taking the continuous limit of the underlying microscopic Ising chain with Kac potentials after a feasible scaling. We indicate reference [1] for details on this procedure, that is known as the Lebowitz-Penrose limit. We drop hereafter the suffix Λ, for the sake of simplicity.
The axiomatic theory provides that the magnetization evolves in time according to a gradient dynamicsṁ where I represents the local current in which χ β (m) = β 1 − m 2 is the mobility coefficient for an Ising spin system. Hence, the stationary problemṁ = 0 reads that is an integro-differential equation in the unknown function m at constant I and given boundary conditions. We call I = jε, j ∈ R a constant, as we expect the current to be of order ε.

Notation
It is worth redefining the convolution kernel in the distributional sense as follows This way we act on functions defined in the interior of the bulk. For any bounded function If not specified, integrals are intended to be performed on Let A m,h the linear operator acting on a bounded function f as follows The action of the n-th power of A m,h on f is explicitly given by (2.14)

Instantons
We briefly recall a fundamental result obtained by De Masi et al. [14][15][16][17]. This regards the free boundary version of the problem. Let be the free energy functional on R defined for functions that belong to the Banach space In this case the flux is null, so that the stationary problem reduces tȯ is a strictly increasing, antisymmetric function which converges exponentially fast to ±m β as x → ±∞.
Hereafter, we callp for any x and y in R. We refer asm ′ to the derivative ofm with respect to x.

UPHILL DIFFUSION
Our main result is , there is ε β > 0 such that for any ε < ε β there are an antisymmetric, continuous function m and a positive constant j that solve (3.1)

Outline of the Proof
For our purposes, it is worth performing the following change of variables: where, explicitly In this position (2.7) becomes, after a straight integration .
Observe that h (0) = 0 if m is odd, so we eventually formulate problem (3.1) as a system of coupled equations: The existence of a solution of problem (3.5) is proved by iteration (Newton's method): we start from a couple (m 0 , h 0 ) and fixed µ and j and define a feasible map (m n , h n ) → (m n+1 , h n+1 ) that converges uniformly to a couple (m, h) that solves (3.5) and satisfies certain boundary conditions m (±ε −1 ) = ±ν, ν = µ in general. Afterwards, we prove that j can be actually tuned in order to cover the whole metastable region, that is for any ν ∈ (m * (β) , m β ) there exists at least one j > 0 such that lim n→∞ m n (x) solves (3.5) with m (ε −1 ) = ν. In this scheme, the choice of m 0 (and h 0 as a function of m 0 ) turns out to be crucial, as we would like to start with a profile that is "almost" a fixed point.
The technical part of the paper is organized as follows: after having established the recursive method and chosen m 0 , we perform in Section 4 some estimates that are needed in the course of the proof; in particular, we prove the invertibility of I − A m,h . In Section 5 we construct the sequence (m n , h n ) ∞ n=0 and prove convergence to a certain solution of (3.5) with j > 0. In Section 6 we deal with the invertibility issue mentioned above.

Choice of m 0
Proposition 3.2. The "macroscopic" problem at β > 1 Proof. We refer to (3.7) as the macroscopic equation because it comes from the variational problem that one obtains after performing the macroscopic limit (see [10,13]). A straight integration gives with j M fixed by the choice of µ − and µ + : As a function of M , x is infinitely times differentiable and moreover, x (M ) is invertible since M ′ is negative. M can be obtained as the unique real solution of the cubic equation (3.8).
Notice that problem (3.7) can be formulated as a system of coupled equations as well For technical reasons, we choose ε −1 so large thatm (ε − 1 2 /2) = m β − δ, δ > 0 a small parameter specified further on. We speculate that if ε −1 is large enough, the solution should not differ so much from the instanton in the nearby of the origin. Once reached the value m 0 (ε − 1 2 ) ≈ m β , we suppose the solution to be monotone decreasing and "close" to the (rescaled) macroscopic profile. This will be very clear a posteriori, as we will show that in fact the distance between m 0 and the stationary solution m is of order ε in the sup norm.

Iterative scheme
The following results explicitly defines the method.
In the iterations, j = j (µ 0 ) is fixed parameter, whose value is actually specified by the boundary value m 0 (ε −1 ) = µ 0 (andm (ε − 1 2 ) that depends on ε only): Every time an iteration is performed, the boundary value changes, and therefore we cannot rule out the possibility that our constructive method defines a map j → (m * (β) , m β ) which is not surjective. Hence, Proposition 3.4 is needed in order to close the proof of Theorem 3.1.

A preliminary result
We recall here a result proved in [23] and that can be even found in [12]. Define the scalar product on R and indicate m ′ :=m ′ / (m ′ ) 2 ∞ . We have the following Proposition 4.1. There are positive constants a and c such that for any f ∈ L ∞ (R) and any integer n: There is a very straight consequence of this result, which is however essential for our purposes.  Proof. Since m ′ is symmetric, ψ m ′ ∞ = 0 and then ψ ≡ ψ.

Invertibility of I − A m,h
Define the set: .
Proof. It suffices to show that for any integer n, A n m,h ψ ε ≤ γ n ψ ε . If n ≤ n ε this is true because of (4.14). If n > n ε , we write n = k n n ε + m n , with m < n ε so that 4.18) and notice that Ψ (k) is antisymmetric and satisfies Ψ (k) ε ≤ γ mn ψ ε by virtue of (4.14). Thus, for any i < k: is antisymmetric and therefore by iteration Summing on k we get (4.16).
The previous bound induces the existence of the inverse of I − A m,h . Explicitly Lemma 4.7. Let m ∈ Σ δ,ε , h = H (m). For any bounded, antisymmetric function F on can be solved in the unknown function ϕ. Furthermore, Proof. It is a straightforward consequence of Proposition 4.6.

Small perturbations to m 0
In this section we construct m 1 as a series in which each correction ϕ n depends on the previous ones with ϕ 0 ≡ 0. For notational convenience, we will often indicate We restrict to the positive semiline. We have, for any 0 ≤ x ≤ ε − 1 2 − 1: In the interval ε − 1 2 + 1 ≤ x ≤ ε −1 , similarly to estimate (4.15): where we used Lagrange's Theorem. The remaining case is when where we bounded the first term with the sup norm of the derivative ofM µ , while the second term is exponentially small in by virtue of the previous estimates. This proves that S 0 ε = O (ε). Indeed, the existence of ϕ 1 follows from the invertibility of I − A m 0 ,h 0 , and explicitly: for some t n ∈ R; (ii) there is a constant τ such that ϕ n ε ≤ τ ϕ n−1 2 ε for any n ≥ 2; (iii) lim n→∞ m 1 − m 0 − φ n ε = 0, where m 1 is a solution of (3.12) at n = 1; Proof. It works by induction. In particular, suppose that (i) and (ii) hold for any integer less or equal to a certain k. Since ϕ 1 ε ≤ c 0 ε, iterating (ii) we get and then Moreover if ε is so small that m 0 + φ k ∈ Σ δ,ε , and then A m 0 +φ k ,h 0 is invertible, and ϕ k+1 exists. We prove (ii); expand the hyperbolic tangent in Taylor series: Combining (5.14) with the definition of ϕ n we get so we can identify τ ≡ β/1 − γ. This proves that for any integer n: Taking the limit n → ∞, by continuity of the hyperbolic tangent, we get the uniform convergence to m 1 . (iv) follows from (5.13).

Small perturbations to h 0
(i) there exists a continuous, antisymmetric function m solution of Proof. We construct m using again Newton's method with starting point (m 0 , h). Observe that hence, if δ ′ < δ/4β 2 we are in the hypothesis of Lemma 4.7, so we conclude that there exists an antisymmetric function ψ 1 that solves for some s 1 ∈ R, and moreover ψ 1 ε ≤ τ h − h 0 ε . The rest of the proof is the same as that of Proposition 5.2, provided δ ′ ≤ δ/ (2τ + 1), which is the condition needed in order to apply Lemma 4.7 recursively.

Further corrections
As we shall see, it is worth emphasizing scaling properties of the magnetization profile by introducing the following weighted norm at fixed α > 0: Notice that convergence in the α-norm implies uniform convergence as the inclusion · ε,α ≤ · ε ≤ e α · ε,α holds. The iterability of our method directly follows from the fact that (m n , h n ) → (m n+1 , h n+1 ) is a contraction in the α-norm (for a feasible choice of parameters).
Proof. m (1) and m (2) exist by Proposition 5.3. By Taylor's Theorem p 1,2 being some interpolating function between p m (2) ,h (2) and p m (1) ,h (1) . We multiply by e −αε|x| equation (5.24) and take absolute values to get (5.26) as an equality can be solved in the unknown function e −αε|x| |m (2) In this case, i.e. if ε is small enough:

Convergence to (m, h)
We now prove Proposition 3.3.
Proof. Suppose that for any k < n, n a fixed integer, the following hypothesis hold true: (H1) there is a continuous, antisymmetric function m k which solves (H2) there is a constant ρ ∈ (0, 1) independent of k such that Notice that (H1) and (H2) imply: According to (5.30), α and ε can be suitably tuned in order to apply Lemma 5.4; in particular, this is true uniformly in k provided h 1 − h 0 ε ≤ (1 − ρ) /e α ρ. In this hypothesis there exists m n solution of satisfying a certain boundary condition. It remains to prove that (H2) holds for k = n. We have, for any x ∈ [−ε −1 , ε −1 ]: Hence, in the α-norm: If ρ ≤ 2βc τ /α and ε is accordingly small, (m n , h n ) → (m n+1 , h n+1 ) is a contraction in the α-norm. This implies uniform convergence to a solution of (3.5).