Forward-backward correspondence between stationary structure and splitting probabilities in active matter

Technical Summary: Forward–Backward Correspondence in Active Matter

Problem Statement
Active particles confined by hard walls exhibit two distinct phenomena driven by directional persistence in the absence of thermal fluctuations: (1) dynamical adsorption, where particles accumulate at boundaries, and (2) a finite wall splitting probability ($\pi_L(0) > 0$), meaning a particle initialized at a wall has a non-zero probability of reaching the opposite boundary before returning. While stationary distributions are governed by the forward Kolmogorov equation and first-passage observables (like splitting probabilities) by the backward equation, the relationship between these two descriptions has remained unclear. Previous work established this connection for discontinuous random walks (jumping particles), but it was unknown whether continuous active models (Run-and-Tumble, Active Brownian, and Active Ornstein–Uhlenbeck particles) share this property.

Methodology
The author formulates the problem in the extended position–velocity phase space $(x, v)$, treating the three canonical active matter models (RTP, ABP, AOUP) on a common footing. The methodology involves:

1. Microscopic Analysis: Demonstrating that finite splitting probabilities arise from ballistic short-time dynamics in the zero-temperature limit, where particles move away from a wall before stochastic reorientation occurs.
2. Operator Comparison: Deriving the stationary forward equation for the bulk distribution $n(z, w)$ and the backward equation for the splitting probability $\pi_\lambda(z, w)$.
3. Auxiliary System Construction: Introducing an auxiliary forward density $\tilde{n}(z, w)$ where the drift term is reversed ($w \to -w$). This system is identified as the Siegmund dual of the original system.
4. Boundary Matching: Comparing the boundary conditions of the auxiliary density $\tilde{n}(z, w)$ with the spatial derivative of the splitting probability, $u(z, w) = \partial_z \pi_\lambda(z, w)$. By showing that these two quantities satisfy the same differential equation and boundary conditions (up to a proportionality constant), the author establishes an exact mathematical correspondence.

Key Contributions and Results
The paper derives exact identities linking the stationary distribution and splitting probabilities for RTP, ABP, and AOUP dynamics in arbitrary spatial dimensions $d$. The core results are:

• Velocity-Resolved Correspondence:
  $$p_v(w) \partial_z \pi_\lambda(z, w) = \rho(z, -w)$$
  $$p_v(w) \pi_\lambda(0, w) = f_0(-w)$$
  Here, $\rho(z, w)$ is the stationary bulk density, $p_v(w)$ is the stationary velocity distribution, $\pi_\lambda(z, w)$ is the splitting probability, and $f_0(w)$ is the velocity distribution of particles adsorbed at the left wall.

• Marginal Correspondence:
  Integrating over velocity yields the central finding:
  $$\partial_z \pi_\lambda(z) = \rho(z)$$
  $$\pi_\lambda(0) = f_w$$
  This states that the fraction of particles dynamically adsorbed at a wall ($f_w$) is exactly equal to the probability that a particle starting at that wall reaches the opposite boundary ($\pi_L(0)$).

• Generalization to Jumping Processes:
  The framework is extended to generalized RTP models with arbitrary velocity distributions. This connects the stationary adsorption fraction to the jump-length distribution, allowing the translation of first-passage results for Lévy-flight-like processes (specifically those with algebraic velocity tails) into exact predictions for stationary adsorption fractions.

• Numerical Verification:
  The theoretical identities are verified via numerical simulations for RTP, ABP, and AOUP particles in three dimensions. The results confirm that $f_w$ and $\pi_\lambda(0)$ are indistinguishable within statistical error and exhibit the same asymptotic scaling ($\propto \lambda^{-1}$) for large separations.

Significance
The paper claims to establish an "exact bridge" between stationary and first-passage descriptions of confined active matter. The significance lies in revealing that dynamical adsorption and finite wall splitting probabilities are not merely related phenomena but are complementary representations of the same persistence-driven dynamics. By comparing forward and backward operators directly, the work provides a unified framework that applies to continuous active models, resolving the question of whether the $\pi_L(0) > 0$ condition is exclusive to discontinuous random walks. The correspondence allows for the direct conversion of known first-passage results into exact predictions for stationary structural properties, such as the density of adsorbed particles.