A topological fluctuation theorem

Fluctuation theorems specify the non-zero probability to observe negative entropy production, contrary to a naive expectation from the second law of thermodynamics. For closed particle trajectories in a fluid, Stokes theorem can be used to give a geometric characterization of the entropy production. Building on this picture, we formulate a topological fluctuation theorem that depends only by the winding number around each vortex core and is insensitive to other aspects of the force. The probability is robust to local deformations of the particle trajectory, reminiscent of topologically protected modes in various classical and quantum systems. We demonstrate that entropy production is quantized in these strongly fluctuating systems, and it is controlled by a topological invariant. We demonstrate that the theorem holds even when the probability distributions are non-Gaussian functions of the generated heat.


I. THE EXACT WINDING NUMBER DISTRIBUTION FOR A SINGLE VORTEX.
Here we provide the derivation of the exact winding number distribution in the case where the force-field is generated by a single vortex line. The Fokker-Planck equation associated to the dynamics of a stochastic particle in the vortex field defined by Eq. (11) of the main text is given by ∂ t P(r, φ, z, t) + γD 2πr 2 ∂ φ P(r, φ, z, t) = D ∂ 2 rr + 1 r ∂ r + 1 where at t = 0 the distribution P(r, φ, z, t) satisfies P(r, φ, z, 0) = r −1 δ(r − r 0 )δ(φ − φ 0 )δ(z − z 0 ) with r 0 > 0, while φ 0 and z 0 are set to 0 without loss of generality. Due to the translational invariance of the problem along the z axis, solutions of the Fokker-Planck equation [Eq. (S1)] are written as where P 2D (r, φ, t) can itself be decomposed into separable functions of the form e iuφ e −Dλ 2 t ρ(r) [1]. Meanwhile, ρ(r) satisfies For what follows, we shall consider solutions for which Re (k u ) ≥ 0. Solutions of Eq. (S3) take the general form where J ν and Y ν are Bessel functions of the first and second kind respectively, of order ν. Moreover, since for a vortex-generated field in open space the Brownian particle avoids the origin with probability 1 [2], the distribution satisfies at all times, angles and z, P(0, φ, z, t) = P(r → +∞, φ, z, t) = 0. From this constraint, C Y can be set to 0 in what follows. The distribution then becomes In order to satisfy the initial condition P 2D (r, φ, 0) = r −1 δ(r − r 0 )δ(φ), we use a closure relation for Bessel functions To simplify this further, we use the following relation valid for α, β > 0, Re(ν) > −1, and where I ν is the modified Bessel function of the first kind of order ν. This gives us where I ν is the modified Bessel function of the first kind, of order ν. For closed trajectories, the winding number distribution associated with a particular initial position (r 0 , φ 0 = 0, z 0 = 0) is p(n, t|r 0 ) ≡ 1 N P(r 0 , φ = 2πn, 0, t) with N ≡ +∞ n=−∞ p(n, t|r 0 ), giving Eq. (12) of the main text. * ramin.golestanian@ds.mpg.de

II. TOPOLOGICAL PHASES IN ONE DIMENSION
Here we consider a one dimensional variant of the model studied in the main text and show that it allows to derive similar results. For simplicity, we examine a dynamics with two internal states A and B and where the possible transitions at site x read with a denoting the lattice step while γ ext , γ ext and γ in are the rates between sites and internal states, respectively. The topology described by the transitions in Eq. (S9) is similar, for instance, to that studied in Refs. [4][5][6]. Denoting ρ A,B (x, t) the densities at site x associated with states A and B at time t, they obey the following master equations: (S10b) Following the calculation steps described in Methods, we now define as respectively the total density and the difference between the two states occupations at site x. Considering the It is clear from Eqs. (S12) that, contrary to δ, ρ is slow as it is conserved. Therefore, in the long-time limit and for slowly varying fields δ can be enslaved to ρ. Setting ∂ t δ = 0, solving Eq. (S12b) recursively and replacing the expression of δ so obtained we find that at order a 2 Eq. (S12a) takes the simple drift-diffusion form with effective drift and diffusivity Eq. (S13) thus maps the lattice model (S9) to the stochastic motion of a particle in one dimension with constant drift v and diffusivity D. Considering the model on a ring, i.e. with periodic boundary conditions, the topological fluctuation theorem [Eq. (8) of the main text] predicts for the winding number probability ratio with N the total number of lattice sites. As for the two dimensional system, the rhs of Eq. (S14) takes a system size dependency which here can however be compensated by fine tuning the amplitude of the rates γ ext and γ ext .
Considering now an open system, the stationary solution of Eq. (S13) corresponds to a density profile exponentially localized at the right (resp. left) edge of the system for v > 0 (resp. v < 0) over a characteristic lengthscale

III. COARSE GRAINING OF THE LATTICE MODEL WITH SYMMETRIC RATES
Here we sketch the derivation of Eq. (15) of the main text for the case where the lattice model of Fig. 4 includes symmetric transition rates. The derivation follows similar lines as the one presented in Methods.
The bulk dynamics Let us first recall the transition rules of the lattice model. Considering a site at position (x, y) the allowed transitions are: where the letters A-D label the four internal states and a is the lattice spacing. In Eq. (S15) γ ext and γ ext denote the external clockwise and counter-clockwise transition rates, while γ in and γ in are the counter-clockwise and clockwise internal rates. Taking into account all possible transitions and neglecting the influence of the system's boundary, the master equations governing the bulk dynamics read where, as in the main text, we have used the 4-periodic map σ: i ∈ {0, ..., 3} → {A,...,D}, as well as the notations x = xê x + yê y and ∆x i = aR(− iπ 2 )ê x . Defining the following fields and taking the continuum limit Eqs. (S16) are recast up to order a 2 as where we have used the following definitions for the spatial derivative operators As for the fully chiral case, ρ b is the only slow field in Eq. (S17) and we now enslave ρ 1,2,3 . Keeping terms up to order a 2 , we find that ρ 1 = O(a) can be neglected, while with α ≡ γ ext + γ ext + γ in + γ in and β ≡ γ ext − γ ext − (γ in − γ in ). Solving the above equation for ρ 2,3 and replacing the solution into Eq. (S17a) we recover (S19) As for the fully chiral case presented in Methods, we find that the bulk dynamics is fully diffusive with an isotropic bulk diffusivity symmetric by exchange of indices int and ext, while its limiting behaviors read so that it always remains finite even for fast external rates.
The dynamics near boundaries We now turn to the characterization of the dynamics at the lower edge of the system, which we couple to the bulk density ρ b which plays the role of the density at sites A and D. The corresponding master equations read Similarly to the fully chiral case, we define Taking the continuous limit and expanding Eqs. (S20) up to second order in a, we find after some algebra Solving the above two last equations for ρ 1,2 at zeroth order in a, we get the lengthly expressions which imply the following relative occupations for the boundary states in the limit of vanishing gradients .
Using the definition of ρ e , we thus obtain the ratio of edge to bulk densities whose expression for γ in , γ in γ ext , γ ext is reported in Eq. (16) of the main text. In the opposite limit of fast internal rates, Eq. (S22) reads where we have used the definitions ζ in = γ in /γ in and ζ ext = γ ext /γ ext . In contrast with the fast external rate limit, Eq. (S23) implies no significant density accumulation at the system boundary.
In the limit of strong chirality ζ ext , ζ in → 0, expressing these coefficients at leading order in ζ ext and ζ ext leads to the expressions given in the main text.