Local non-equilibrium thermodynamics

Local Shannon entropy lies at the heart of modern thermodynamics, with much discussion of trajectory-dependent entropy production. When taken at both boundaries of a process in phase space, it reproduces the second law of thermodynamics over a finite time interval for small scale systems. However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally. Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials. We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation.

A colloidal particle is released at t = − for a small > 0 from the upper-left corner so that it has non-vanishing probability at t = 0, and moves stochastically. We repeat the experiment, and consider to block the particle by a partition incompletely as shown in the right panel. During non-equilibrium evolution, the presence of the wall increases the information content of the ensemble Λ A of A resulting in an increase of the local free energy ψ(A, t) for some t. Accordingly, ψ(B, t) would be decreased.

B. Derivation of Eq. (9).
Let Λ be the set of all possible space-time trajectories of an experiment. When we count the accessible number of paths, we assign each path a weight of the form g(l) ≡ e −s(l) , where s(l) is such that the probability of path l is represented as p(l) ∝ e −s(l) , so that less probable paths are to be less counted. Then, a probability of path l in a sample space X would be p(l) = g(l)/ l∈X g(l). It would be convenient to deal with a trajectory in a discrete approximation as a set of states x i in consecutive times t i for i = 0, · · · , n. Let us denote the set of trajectories that pass x k at time t k as Λ(x k , t k ) for some integer k (see Fig. S2). We will calculate the ratio of the number of paths in Λ(x k , t k ) to the number of all possible trajectories in Λ. Since the probability of path l is p(l) = g(l)/ l∈Λ g(l), the ratio becomes l∈Λ(x k ,t k ) p(l). Then, the ratio would be approximated as x 0 · · · x k · · · xn p(x 0 , · · · , x k , · · · , x n ), where each sum is taken over the phase-space points at time t i , and x k means that we omit the sum over the phasespace points at time t k . It is important to omit that sum at time t k , otherwise the result would be 1. By the law of total probability the sum reduces to p(x k , t k ).

C. Derivation of dynamic rules.
We consider the Langevin equation, which is a minimal prototype that contains an essence of stochasticity: ζẋ = −∇E(x, t) + ξ, where E(x, t) is energy of a microstate x at time t, ζ is the friction constant and ξ is the fluctuating force that satisfies the fluctuation-dissipation theorem, ξ(t)ξ(t ) = 2k B T ζδ(t − t ). The probability density p(x, t) of a system under the overdamped Langevin equation obeys the Fokker-Planck equation as From p(x, t) = e φ(x,t)−φ 0 (see the main text or Sections A and B above), we have ∂p ∂x = p ∂φ ∂x and ∂ 2 p ∂x 2 = p ∂φ ∂x 2 + p ∂ 2 φ ∂x 2 . Substituting these relations into the Fokker-Planck equation, and dividing by p(x, t), we obtain the following equation: Here the first two terms in the right-hand side may be written as ∂ ∂x (E(x, t) + k B T φ(x, t)) ∂φ(x,t) ∂x , and the second two terms as . Noting that the sum in the paren-thesis is just ψ(x, t) (see the main text or Section A above), we have where ∇ 2 denotes the Laplacian. This is a non-linear convection-diffusion equation with a source term. It determines the dynamics of φ and ψ completely, given initial and boundary conditions, where the energetic cost of the information flow is mediated by heat with less than 100% efficiency resulting in a net loss of information, andthus a net increase of entropy.

D. Details of in-silico experiments.
We carried out in-silico experiments of the Brownian movement of a particle. We numerically solve the following overdamped Langevin equation: where E(x, t) is energy of a microstate x, and thermal fluctuation ξ satisfies ξ(t)ξ(t ) = 2k B T ζδ(t − t ). Here we set ζ = 1 for simplicity. Then, after discretization, we have where = t i+1 − t i and r(t i ) is a random number drawn from the standard normal distribution. In our simulations, we set the mobility to unity, and k B T to 2 by rescaling. We constrained the particle by setting the reflecting boundaries at x = 0 and x = L (L = 10).
We used = 0.01. The domain is partitioned into 50 bins, and we counted the number of particles for each bin at each time to obtain the graphs of information and free energy. Figure S3 shows the profiles of information and free energy when the initial condition is set to p(x, 0) = δ(x − L/4). In this case, there is no barrier in ψ(x, t) although there is an energy barrier. Thus, the local equilibrium is established quickly in the initial stage.
Then the free energy ψ drives information φ to the right region until reaching the global equilibrium. During this second stage, the flow continues without breaking the established local equilibrium. Note that the profile of information evolves from the hat shape to the shape that exactly compensates the energy profile up to an additive constant.
Now we put initially the particle at the location of the global minimum of energy, i.e.
p(x, 0) = δ(x − 3L/4), and Fig. S4 shows the profiles of information and free energy over microstates. There is no barrier in the free energy profile ψ. First, the local equilibrium is established quickly, and the flow of information φ continues towards the left region although the speed is much slower than the local equilibration process. Second, the flow of information φ continues until the global equilibrium is established without breaking the local equilibrium.
In this case again, the profile of information at the equilibrium compensates exactly the energy profile up to an additive constant. for the left figures, and t 101 to t 4000 for the right figures. Due to the initial condition, there is no free energy barrier in this case. The process towards local equilibrium is very quick. The global equilibrium proceeds without breaking the established local equilibrium.