arXiv subjects

Derek Frydel

Publications and source records attributed to Derek Frydel.

At least 19 recordsLinked to original sources

arXiv2026

Semi-Markovian switching in a fluctuating harmonic trap: An age-structured formulation

We study a Brownian particle in a harmonic trap whose stiffness switches between two values with arbitrary waiting-time statistics, generating semi-Markovian dynamics. To treat the resulting temporal memory, we formulate the problem in an enlarged age-structured state space, restoring Markovianity and yielding a local Fokker--Planck description. Within this framework, we derive exact steady-state integral equations for the spatial and birth distributions and obtain exact expressions for stationary moments, injected power, and potential energy. In the second part of the paper, we analyze the stochastic-resetting limit, corresponding to a particle alternately released and trapped. By representing the stationary spatial distribution as a superposition of Gaussian states with fluctuating variance, the problem can be reformulated as a switching process in variance space. This yields exact integral equations for the variance distributions and leads to a simplified description amenable to direct analytical treatment.

cond-mat.stat-mech↗

arXiv2026

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

Active particles confined by hard walls accumulate at boundaries and may become dynamically adsorbed due to directional persistence. In this work, we show that the same persistence mechanism also gives rise to a finite wall splitting probability, meaning that a particle initialized at a wall can reach the opposite boundary before returning to its starting point. By comparing forward and backward evolution equations directly in position--velocity phase space, we derive exact relations linking stationary distributions and splitting probabilities for run-and-tumble, active Brownian, and active Ornstein--Uhlenbeck particles. In particular, we show that the stationary density is generated by the spatial derivative of the splitting probability, while the distribution of dynamically adsorbed particles at the walls is encoded in wall splitting probabilities. The correspondence is valid in arbitrary spatial dimension and establishes an exact bridge between stationary and first-passage descriptions of confined active matter, revealing them as complementary representations of the same persistence-driven dynamics.

cond-mat.stat-mech↗

arXiv2025

Search by Return: Stochastic Resetting in Fluctuating Harmonic Potentials

We study a class of stochastic resetting (SR) processes in which a diffusing particle alternates between free motion and confinement by an externally controlled potential. When the particle is recaptured, it undergoes a return trajectory that drives it toward a designated reset point. In standard SR, such returns are treated as instantaneous, but in realistic setups they have finite duration and introduce imprecision in the starting points of subsequent search attempts. We analyze a fluctuating harmonic potential in which return trajectories are forcibly terminated the moment the particle reaches the origin, ensuring that all outward (diffusive) trajectories begin from the same point. This is implemented through instantaneous positional information: a feedback signal that shortens the return phase without incurring additional mechanical energetic cost. We examine several search protocols built on this controlled return mechanism and determine their mean first-passage times (MFPTs). Of particular interest is a protocol in which outward diffusion is eliminated entirely and the return motion itself becomes the search mechanism. This "search by return" perspective reverses the conventional logic of SR and yields a closed-form MFPT that highlights the efficiency of using return dynamics as the primary search strategy.

cond-mat.stat-mech↗

arXiv2025

Integral equation formulation of run-and-tumble particles in a harmonic trap: the special status of a system in two-dimensions

Statistical-mechanical models often exhibit a dimension-dependent solvability: in 1D, exact solutions are straightforward; in 2D, solutions are exact but require nontrivial derivations; and in 3D, closed-form solutions are typically unavailable. This logic is repeated for a simple model of self-propelled particles, run-and-tumble particles (RTP) in a harmonic trap, confirming the claim that the system in 2D enjoys special status. This study revisits the RTP-harmonic-trap model using an integral-equation formulation recently proposed in Ref. \cite{POF-Frydel-2024}. The formulation is based on reinterpreting RTP motion as a jump process. The key quantity of the formulation is a transition operator $G(x,x')$, representing the probability distribution of the jumps of an auxiliary system. The stationary distribution is then obtained from the integral equation $\rho(x) = \int dx' \, \rho(x') G(x,x')$. In 2D, we find that $G(x,x')$ is reversible. This implies the $\rho(x)$ satisfied the detailed balance condition, $\rho(x') G(x,x') = \rho(x) G(x',x)$, from which $\rho(x)$ can be obtained without need of an integral equation. The reversibility of $G(x,x')$ does not mean that RTP particles are in equilibrium. It only means that our specific interpretation of the RTP motion leads to an auxiliary system that is in equilibrium. The reversibility of the system in 2D is lost if the probability distribution of the waiting times (the times that determine the duration on the "run" stage of the RTP motion) deviates from an exponential distribution.

cond-mat.stat-mech↗

arXiv2024

Run-and-tumble particles in slit geometry as a splitting probability problem

Run-and-tumble particles confined between two walls seem like a simple enough problem to possess analytical tractability. Yet up to date, no satisfactory analysis is available for dimensions higher than one. This work contributes to the theoretical understanding of this system by reinterpreting it as a splitting probability problem. Such reinterpretation permits us to formulate the problem as the integral equation, rather than a more standard differential equation based on the Fokker-Planck equation. In addition to providing an analogy with another phenomenon, the reinterpretation permits a new type of analysis, yields useful results, and offers some analytical tractability.

cond-mat.stat-mech↗

arXiv2024

Statistical mechanics of passive Brownian particles in a fluctuating harmonic trap

We consider passive Brownian particles trapped in an "imperfect" harmonic trap. The trap is imperfect because it is randomly turned off and on, and as a result, particles fail to equilibrate. Another way to think about this is to say that a harmonic trap is time-dependent on account of its strength evolving stochastically in time. Particles in such a system are passive and activity arises through external control of a trapping potential, thus, no internal energy is used to power particle motion. A stationary Fokker-Planck equation of this system can be represented as a third-order differential equation, and its solution, a stationary distribution, can be represented as a superposition of Gaussian distributions for different strengths of a harmonic trap. This permits us to interpret a stationary system as a system in equilibrium with quenched disorder.

cond-mat.stat-mech↗

arXiv2023

Active oscillator: recurrence relation approach

The present work analyzes stationary distributions of active Brownian particles in a harmonic trap. Generally, obtaining stationary distributions for this system is non-trivial, and up to date no exact expressions are available. In this work, we develop and explore a method based on a transformation of the Fokker-Planck equation into a recurrence relation for generating moments of a distribution. The method, therefore, offers an analytically tractable approach, an alternative to numerical simulations, in a situation where more direct analytical approaches fail. Although the current work focuses on the active Brownian particle model, the method is general and valid for any type of active dynamics and any system dimension.

cond-mat.stat-mech↗

arXiv2023

Run-and-tumble oscillator: moment analysis of stationary distributions

When it comes to active particles, even an ideal-gas model in a harmonic potential poses a mathematical challenge. An exception is a run-and-tumble model (RTP) in one-dimension for which a stationary distribution is known exactly. The case of two-dimensions is more complex but the solution is possible. Incidentally, in both dimensions the stationary distributions correspond to a beta function. In three-dimensions, a stationary distribution is not known but simulations indicate that it does not have a beta function form. The current work focuses on the three-dimensional RTP model in a harmonic trap. The main result of this study is the derivation of the recurrence relation for generating moments of a stationary distribution. These moments are then used to recover a stationary distribution using the Fourier-Lagrange expansion.

cond-mat.stat-mech↗

arXiv2022

Entropy production of active particles in underdamped regime

The present work investigates the effect of inertia on the entropy production rate $\Pi$ for all canonical models of active particles for different dimensions and the type of confinement. To calculate $\Pi$, the link between the entropy production and dissipation of heat rate is explored resulting in a simple and intuitive expression. By analyzing the Kramers equation, alternative formulations of $\Pi$ are obtained and the virial theorem for active particles is derived. Exact results are obtained for particles in an unconfined environment and in a harmonic trap. In both cases, $\Pi$ is independent of temperature. For the case of a harmonic trap, $\Pi$ attains a maximal value for $\tau = \omega^{-1}$ where $\tau$ is the persistence time and $\omega$ is the natural frequency of an oscillator. For active particles in 1D box, or other non-harmonic potentials, thermal fluctuations are found to reduce $\Pi$.

cond-mat.stat-mech↗

arXiv2022

Theory of Charge Regulation of Colloidal Particles in Electrolyte Solutions

We present a theory that enables us to calculate the effective surface charge of colloidal particles and to efficiently obtain titration curves for different salt concentrations. The theory accounts for the shift of pH of solution due to the presence of 1:1 electrolyte. It also accounts self-consistently for the electrostatic potential produced by the deprotonated surface groups. To examine the accuracy of the theory we have performed extensive reactive Monte Carlo simulations, which show excellent agreement between theory and simulations without any adjustable parameters.

cond-mat.soft↗

arXiv2022

Positing the problem of stationary distributions of active particles as third-order differential equation

In this work, we obtain third order linear differential equation for stationary distributions of run-and-tumble particles in two-dimensions in a harmonic trap. The equation represents the condition $j = 0$ where $j$ is a flux and is obtained from inference, using different known results in the limiting conditions. Since the analogous equation for passive Brownian particles is first order, a second and third order term must be a feature of active motion. In addition to formulating the problem as third order equation, we obtain solutions in the form of convolution of two distributions, the Gaussian distribution due to thermal fluctuations, and the beta distribution due to active motion at zero temperature. The convolution form of the solution indicates that the two random processes are independent and the total distribution is the sum of those two processes.

cond-mat.stat-mech↗

arXiv2021

Kuramoto model with run-and-tumble dynamics

This work considers an extension of the Kuramoto model with run-and-tumble dynamics -- a type of self-propelled motion. The difference between the extended and the original model is that in the extended version angular velocity of individual particles is no longer fixed but can change sporadically with a new velocity drawn from a distribution g(w). Because the Kuramoto model undergoes a phase transition, it offers a simple case study for investigating phase transition for a system with self-propelled particles.

cond-mat.stat-mech↗

arXiv2021

Reactive Monte Carlo Simulations for Charge Regulation of Colloidal Particles

We use a reactive Monte Carlo simulation method and primitive model of electrolyte to study acid-base equilibrium that controls charge regulation in colloidal systems. The simulations are performed in a semi-grand canonical ensemble in which colloidal suspension is in contact with a reservoir of salt and strong acid. The interior of colloidal particles is modeled as a low dielectric medium, different from the surrounding water. The effective colloidal charge is calculated for different number of surface acidic groups, pH, salt concentrations, and types of electrolyte. In the case of potassium chloride the titration curves are compared with the the experimental measurements obtained using potentiometric titration. A good agreement is found between simulations and experiments. In the case of lithium chloride specific ionic adsorption is taken into account through partial dehydration of lithium ion.

cond-mat.soft↗

arXiv2021

Stationary distributions of propelled particles as a system with quenched disorder

This article is the exploration of the viewpoint within which propelled particles in a steady-state are regarded as a system with quenched disorder. The analogy is exact when the rate of the drift orientation vanishes and the linear potential, representing the drift, becomes part of an external potential, resulting in the effective potential $u_{eff}$. The stationary distribution is then calculated as a disorder-averaged quantity by considering all contributing drift orientations. To extend this viewpoint to the case when a drift orientation evolves in time, we reformulate the relevant Fokker-Planck equation as a self-consistent relation. One interesting aspect of this formulation is that it is represented in terms of the Boltzmann factor $e^{-\beta u_{eff}}$. In the case of a run-and-tumble model, the formulation reveals an effective interaction between particles.

cond-mat.stat-mech↗

arXiv2020

The presence of non-analyticities and singularities in the wavefunction and the role of invisible delta potentials

This article examines the suggestion made in Ref. [EPL, 115 (2016) 60001] that a solution to a particle in an infinite spherical well model, if it is square-integrable, is a physically valid solution, even if at the precise location of the singularity there is no underlying physical cause, therefore, the divergence would have to be a nonlocal phenomenon caused by confining walls at a distance. In this work we examine this claim more carefully. By identifying the correct differential equation for a divergent square-integrable solution and rewriting it in the form of the Schroedinger equation, we infer that the divergent wavefunction would be caused by the potential V(r)=-r delta(r), which is a kind of attractive delta potential. Because of its peculiar form and the fact that it leads to a divergent potential energy = - infinity, the potential V(r) and the divergent wavefunction associated with it are not physically meaningful.

quant-ph↗

arXiv2020

Charge regulation of colloidal particles in aqueous solutions

We study charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. In contact with water these surface groups undergo proton transfer reaction, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. The theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.

cond-mat.soft↗

arXiv2020

Thermodynamic collapse in a lattice-gas model for a two-component system of penetrable particles

We study a lattice-gas model of penetrable particles on a square-lattice substrate with same-site and nearest-neighbor interactions. Penetrability implies that the number of particles occupying a single lattice site is unlimited and the model itself is intended as a simple representation of penetrable particles encountered in realistic soft-matter systems. Our specific focus is on a binary mixture, where particles of the same species repel and those of the opposite species attract each other. As a consequence of penetrability and the unlimited occupation of each site, the system exhibits thermodynamic collapse, which in simulations is manifested by an emergence of extremely dense clusters scattered throughout the system with energy of a cluster $E\propto -n^2$ where $n$ is the number of particles in a cluster. After transforming a particle system into a spin system, in the large density limit the Hamiltonian recovers a simple harmonic form, resulting in the discrete Gaussian model used in the past to model the roughening transition of interfaces. For finite densities, due to the presence of a non-harmonic term, the system is approximated using a variational Gaussian model.

cond-mat.soft↗

arXiv2019

Charge Regulation of Colloidal Particles: Theory and Simulations

To explore charge regulation (CR) in physicochemical and biophysical systems, we present a model of colloidal particles with sticky adsorption sites which account for the formation of covalent bonds between the hydronium ions and the surface functional groups. Using this model and Monte Carlo simulations, we find that the standard Ninham and Parsegian (NP) theory of CR leads to results which deviate significantly from computer simulations. The problem of NP approach is traced back to the use of bulk equilibrium constant to account for surface chemical reactions. To resolve this difficulty we present a new theory of CR. The fundamental ingredient of the new approach is the sticky length, which is non-trivially related with the bulk equilibrium constant. The theory is found to be in excellent agreement with computer simulations, without any adjustable parameters. As an application of the theory we calculate the effective charge of colloidal particles containing carboxyl groups, as a function of pH and salt concentration.

cond-mat.soft↗