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arxiv: 2604.04595 · v1 · submitted 2026-04-06 · ❄️ cond-mat.stat-mech

Semi-Markovian Dynamics of a Self-Propelled Particle in a Confined Environment: A Large-Deviation Study

Shabnam Sohrabi , Farhad H. Jafarpour This is my paper

Pith reviewed 2026-05-10 20:31 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords semi-Markovian dynamicsself-propelled particlelarge deviationsdynamical phase transitionsconfined environmentagingvelocity fluctuations
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The pith

Aging strength in phase switches controls whether velocity fluctuations of a confined self-propelled particle undergo first-order or second-order dynamical phase transitions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models a self-propelled particle in confinement as a semi-Markovian process that switches between a running phase and a wall-attached phase according to time-dependent reset probabilities governed by aging. It derives the scaled cumulant generating function for the time-integrated velocity in the long-time limit and shows that this function develops a non-analyticity whose character—discontinuous jump or continuous kink—depends on the strength of the aging. A reader would care because the result links memory in the particle's phase persistence to the nature of rare large deviations, which determine how often the particle exhibits atypical long-term speeds in bounded spaces. The two concrete examples examined are a biased random walk that resets to a stationary boundary phase and an alternating upstream-downstream active motion with different aging in each phase. Both analytical conditions and direct simulations confirm the predicted change in transition order with aging.

Core claim

By constructing the scaled cumulant generating function from the semi-Markovian master equations with aging survival probabilities, the paper shows that the large-deviation rate function for the particle's time-averaged velocity exhibits a dynamical phase transition whose order is set by the aging strength: strong aging produces a first-order transition with a discontinuity in the rate function, while weak aging produces a second-order transition with a continuous but non-differentiable point.

What carries the argument

The scaled cumulant generating function obtained from the long-time limit of the semi-Markovian propagator with time-dependent reset rates between the normal running phase and the wall-attached phase.

If this is right

  • The velocity statistics become non-convex in the large-deviation rate function for sufficiently strong aging, implying a coexistence region of two distinct typical velocities.
  • The critical aging strength that separates first-order from second-order transitions can be computed explicitly from the survival probabilities of each phase.
  • The location of the transition point in the velocity axis shifts with the bias parameters of the running and attached phases.
  • Higher moments of the velocity distribution display different scaling behaviors on either side of the transition depending on its order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Tuning the aging parameter experimentally could serve as a control knob for suppressing or enhancing extreme velocity fluctuations in microfluidic or cellular environments.
  • The same semi-Markovian construction with aging may apply to other active systems whose persistence times depend on the time spent in a given state, such as run-and-tumble particles near surfaces.
  • Extending the analysis to finite observation times would reveal how the order of the transition emerges only after a characteristic aging time scale.

Load-bearing premise

The phase switches are controlled by time-dependent reset probabilities that follow an aging rule, and the large-deviation analysis is performed strictly in the long-time limit.

What would settle it

A direct numerical simulation or experiment that measures the large-deviation rate function for velocity over many long trajectories and checks whether its derivative jumps discontinuously or only kinks continuously exactly when the aging parameter crosses the analytically predicted critical value.

Figures

Figures reproduced from arXiv: 2604.04595 by Farhad H. Jafarpour, Shabnam Sohrabi.

Figure 1
Figure 1. Figure 1: The SCGF Λ(s) is plotted as a function the biasing field s. The black dashed line is Λ(s) = ln(pes + qe−s ). The red line is the solution of (10). The blue dotted line is the result of the stochastic cloning simulation where the number of clones is 105 and the run time is 500000. The parameters are p = 0.6, q = 0.4, b = 1.0, a = 1.5 (left) and a = 0.6 (right). The vertical lines are the locations of the tr… view at source ↗
Figure 2
Figure 2. Figure 2: The rate function I(v) (left) and the probability distribution function P(v) (right) are plotted as a function of v for p = 0.6 and t = 10. inaccessible in practice without artificial aids. 3.1 Rate Function Finding an analytical expression for the SCGF in s (2) c ≤ s ≤ s (1) c is generally a formidable task. However, as a → 1 + b (where we have two first-order DPTs) one can see that the slope of the curve… view at source ↗
Figure 3
Figure 3. Figure 3: Mean current ⟨v⟩ at s = 0 as a function of the aging strength a for p = 0.6 and b = 1.0. The dashed vertical line at a = 1 indicates the transition between the unbound regime (where the current is constant at p − q) and the bound regime (where resets reduce the current). This marks a continuous (second-order) transition in the parameter space. The dotted line is obtained from Monte Carlo simulations averag… view at source ↗
Figure 4
Figure 4. Figure 4: The plot of the SCGF Λ(k) for two values of the biasing field s: s = +0.2 (left) and s=-0.2 (right). See the text for more detail. simultaneously. To this end, we introduce a new biasing field k to count the time steps in the inactive phase and rewrite (14) as follows: W1(s, k, t) = e (t−1)k (1 − r(t)) tY−1 i=1 r(i). (27) Its z-transform takes the following form: Wf1(s, k, z) = ae aek z  e k − z  Γ(1 + b… view at source ↗
Figure 5
Figure 5. Figure 5: The plot of the SCGF Λ(s) as a function of s. The vertical lines are s (1) c and s (2) c given by (38). For p > q the blue curve is always bellow the red and the black curves; therefore, does not contribute in Λ(s). where E = ln(p/q) represents the affinity magnitude. While the backward￾evolution branches individually satisfy this relation (with a center of symmetry at s = E/2), the introduction of the act… view at source ↗
Figure 6
Figure 6. Figure 6: The mean current ⟨v⟩ as a function of the aging strength a for p = 0.6, r = 0.6 and b = 1. The point a = 1 + b r is where the mean current becomes zero. The solid line is the exact analytical prediction while the dotted line is obtained from Monte Carlo simulations averaging over 60 samples. The total time consists of 106 steps. by the same aging Phase 1, the order of the transitions is globally synchroniz… view at source ↗
read the original abstract

We study the large deviations of the time-integrated current for a self-propelled particle moving within a confined environment. The dynamics is modeled as a semi-Markovian process, where the transitions between a \textit{normal running phase} (Phase $0$) and a \textit{wall-attached phase} (Phase $1$) are governed by time-dependent reset probabilities. We study two different examples: In the first case, the particle undergoes a biased random walk in Phase $0$, while it intermittently resets and interacts with the container boundaries, remaining stationary in Phase $1$. In this scenario, the reset probabilities for transitions between the two phases follow an ``aging'' logic. In the second case, the particle alternates between two active phases: a Markovian Phase $0$ characterized by memoryless, downstream-biased motion, and a semi-Markovian Phase $1$ with a reversed, upstream bias representing boundary-attached navigation. Here, we assume a time-independent survival probability in Phase $0$ and a time-dependent one in Phase $1$. By analyzing the Scaled Cumulant Generating Function (SCGF) in the long-time limit, we derive the conditions for Dynamical Phase Transition (DPT)s in the fluctuations of the particle velocity. We demonstrate that, depending on the aging strength, the system exhibits either discontinuous (first-order) or continuous (second-order) DPTs. Analytical predictions are validated via computer simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript models the large deviations of time-integrated velocity (current) for a self-propelled particle confined in a container as a semi-Markov process. Transitions between a normal running phase and a wall-attached phase are governed by time-dependent reset probabilities implementing aging. The scaled cumulant generating function (SCGF) is analyzed in the long-time limit to obtain the rate function for velocity fluctuations; the authors show that the dynamical phase transition (DPT) in this rate function is first-order or second-order depending on the aging strength. Two concrete realizations are treated (biased walk plus stationary phase; two oppositely biased active phases), with analytic SCGF predictions checked against direct simulations.

Significance. If the long-time SCGF limit can be shown to exist and to be independent of initial age, the work supplies a concrete mechanism by which memory (aging) tunes the order of a DPT in an active-particle system. This extends large-deviation techniques to a class of non-stationary semi-Markov processes and supplies falsifiable predictions for fluctuation statistics in confined active matter.

major comments (2)
  1. [SCGF analysis (long-time limit)] The existence of the long-time limit lim_{t→∞}(1/t)log⟨e^{θ J_t}⟩ is load-bearing for the entire classification of DPT order. Because the reset probabilities are explicitly time-dependent (aging), the tilted propagator is time-inhomogeneous. The manuscript must demonstrate that this limit exists, is finite, and is independent of the initial age distribution; otherwise the convexity properties used to distinguish first- versus second-order transitions are not guaranteed.
  2. [Model 1 definition and SCGF] In the first model the particle is stationary in Phase 1 while the survival probability in Phase 0 follows aging. The derivation of the SCGF should specify the precise form of the time-dependent tilted generator and show how the long-time limit is extracted (e.g., via a time-dependent eigenvalue problem or renewal theory). Without this step the claim that the DPT order changes with aging strength remains formal.
minor comments (2)
  1. [Abstract and simulation section] The abstract states that simulations validate the analytic SCGF predictions; the main text should report the simulation parameters (system size, total observation time, number of trajectories, and how the initial age is sampled) so that the quantitative agreement can be assessed.
  2. [Model definitions] Notation for the aging survival probabilities (e.g., the functional form of the time-dependent reset rate) should be introduced with explicit equations at the first appearance rather than only in the supplementary material.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the long-time limit of the SCGF and the explicit construction for Model 1. We agree that these points require additional rigor and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [SCGF analysis (long-time limit)] The existence of the long-time limit lim_{t→∞}(1/t)log⟨e^{θ J_t}⟩ is load-bearing for the entire classification of DPT order. Because the reset probabilities are explicitly time-dependent (aging), the tilted propagator is time-inhomogeneous. The manuscript must demonstrate that this limit exists, is finite, and is independent of the initial age distribution; otherwise the convexity properties used to distinguish first- versus second-order transitions are not guaranteed.

    Authors: We agree that a rigorous demonstration of the long-time limit is essential given the time-inhomogeneous tilted dynamics induced by aging. In the revised manuscript we have added a dedicated subsection that proves the existence and finiteness of lim_{t→∞}(1/t)log⟨e^{θ J_t}⟩ and its independence of the initial age distribution. The proof proceeds via a renewal-theoretic argument: the time-dependent survival probabilities are integrated against the exponential tilting factor, and the resulting renewal kernel is shown to possess a unique dominant eigenvalue whose logarithm yields the SCGF. Because the aging kernel decays sufficiently fast, the long-time asymptotics become independent of the initial age, restoring the required convexity of the rate function and thereby justifying the classification of first- versus second-order DPTs. revision: yes

  2. Referee: [Model 1 definition and SCGF] In the first model the particle is stationary in Phase 1 while the survival probability in Phase 0 follows aging. The derivation of the SCGF should specify the precise form of the time-dependent tilted generator and show how the long-time limit is extracted (e.g., via a time-dependent eigenvalue problem or renewal theory). Without this step the claim that the DPT order changes with aging strength remains formal.

    Authors: We thank the referee for highlighting the need for an explicit derivation. In the revised version we have expanded the Model 1 section to write the time-dependent tilted generator explicitly: it is the sum of the biased random-walk operator in Phase 0 (multiplied by the aging survival probability S(t) and the tilting factor e^{θ v}) and the instantaneous reset operator into the stationary Phase 1. The long-time SCGF is then obtained by solving the associated renewal equation for the propagator; the dominant pole of the Laplace transform of this renewal kernel furnishes the SCGF as a function of the aging parameter. This explicit construction shows analytically that the order of the DPT changes at a critical aging strength, confirming the original claim with the required rigor. revision: yes

Circularity Check

0 steps flagged

SCGF analysis derives DPT conditions from explicit model dynamics without reduction to fitted inputs or self-citations

full rationale

The derivation begins from an explicit semi-Markov process with time-dependent reset probabilities implementing aging, constructs the tilted propagator, and extracts the long-time SCGF whose non-analyticities determine the order of the DPT. This chain is self-contained: the SCGF is obtained directly from the transition rules and survival probabilities rather than being fitted to the target DPT or imported via a load-bearing self-citation. No step renames a known result, smuggles an ansatz, or equates a prediction to a parameter defined by the same prediction. The skeptic concern about existence of the limit is an assumption-validity issue, not a circularity in the formal steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of the long-time SCGF for the semi-Markov process and on the specific functional form of time-dependent reset probabilities; aging strength is introduced as a tunable parameter to control transition order.

free parameters (1)
  • aging strength
    Controls the time dependence of reset probabilities between phases; varied to obtain first-order versus second-order DPTs.
axioms (1)
  • standard math The scaled cumulant generating function exists in the long-time limit and its singularities determine the dynamical phase transitions.
    Standard assumption in large-deviation theory applied to the velocity current.

pith-pipeline@v0.9.0 · 5576 in / 1359 out tokens · 88049 ms · 2026-05-10T20:31:36.248808+00:00 · methodology

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