Run-and-tumble particles confined between two walls seem like a simple enough problem to possess ... more Run-and-tumble particles confined between two walls seem like a simple enough problem to possess analytical tractability. Yet, 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 formulation 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.
We consider passive Brownian particles trapped in an "imperfect" harmonic trap. The trap is imper... more 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.
The present work analyzes stationary distributions of active Brownian particles in a harmonic tra... more 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 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.
When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathem... more When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathematical challenge. An exception is a runand-tumble particles (RTP) model 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.
The present work investigates the effect of inertia on the entropy production rate for all canoni... more The present work investigates the effect of inertia on the entropy production rate for all canonical models of active particles for different dimensions and the type of confinement. To calculate , 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 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, is independent of temperature. For the case of a harmonic trap, attains a maximal value for τ = ω −1 , where τ is the persistence time and ω is the natural frequency of an oscillator. For active particles in one-dimensional box, or other nonharmonic potentials, thermal fluctuations are found to reduce .
We present a theory that enables us to (i) calculate the effective surface charge of colloidal pa... more We present a theory that enables us to (i) calculate the effective surface charge of colloidal particles and (ii) 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.
In this work, we obtain a third-order linear differential equation for stationary distributions o... more In this work, we obtain a 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. Since an analogous equation for passive Brownian particles is first-order, a second-and third-order term are features of active motion. In all cases, the solution has a form of a convolution of two distributions: the Gaussian distribution representing the Boltzmann distribution of passive particles, and the beta distribution representing active motion at zero temperature.
Journal of Statistical Mechanics: Theory and Experiment, 2021
In this work, we investigate extensions of the run-and-tumble particle model in 1D. In its simple... more In this work, we investigate extensions of the run-and-tumble particle model in 1D. In its simplest version, particle drift is limited to two velocities v = ±v 0 and the model is exactly solvable. Extension of the model to three drifts v = 0, ±v 0 yields the exact solution but the complexity of the expressions indicates that analytical treatment of higher-state models under the same procedure is impractical. Consequently, we modify our goal and consider a generalized version of the model for an arbitrary distribution of states P(v). To analyze such a system, we reformulate the Fokker–Planck equation as a self-consistent relation. The self-consistent relation is then analyzed by means of Laplace transform techniques.
In this work, we consider a lattice-gas model of charge regulation with electrostatic interaction... more In this work, we consider a lattice-gas model of charge regulation with electrostatic interactions within the Debye-Hückel level of approximation. In addition to long-range electrostatic interactions, the model incorporates the nearest-neighbor interactions for representing non-electrostatic forces between adsorbed ions. The Frumkin-Fowler-Guggenheim isotherm obtained from the mean-field analysis accurately reproduces the simulation data points.
This article examines the suggestion made in Ref. [EPL, 115 (2016) 60001] that a solution to a pa... more 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.
Run-and-tumble particles confined between two walls seem like a simple enough problem to possess ... more Run-and-tumble particles confined between two walls seem like a simple enough problem to possess analytical tractability. Yet, 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 formulation 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.
We consider passive Brownian particles trapped in an "imperfect" harmonic trap. The trap is imper... more 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.
The present work analyzes stationary distributions of active Brownian particles in a harmonic tra... more 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 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.
When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathem... more When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathematical challenge. An exception is a runand-tumble particles (RTP) model 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.
The present work investigates the effect of inertia on the entropy production rate for all canoni... more The present work investigates the effect of inertia on the entropy production rate for all canonical models of active particles for different dimensions and the type of confinement. To calculate , 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 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, is independent of temperature. For the case of a harmonic trap, attains a maximal value for τ = ω −1 , where τ is the persistence time and ω is the natural frequency of an oscillator. For active particles in one-dimensional box, or other nonharmonic potentials, thermal fluctuations are found to reduce .
We present a theory that enables us to (i) calculate the effective surface charge of colloidal pa... more We present a theory that enables us to (i) calculate the effective surface charge of colloidal particles and (ii) 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.
In this work, we obtain a third-order linear differential equation for stationary distributions o... more In this work, we obtain a 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. Since an analogous equation for passive Brownian particles is first-order, a second-and third-order term are features of active motion. In all cases, the solution has a form of a convolution of two distributions: the Gaussian distribution representing the Boltzmann distribution of passive particles, and the beta distribution representing active motion at zero temperature.
Journal of Statistical Mechanics: Theory and Experiment, 2021
In this work, we investigate extensions of the run-and-tumble particle model in 1D. In its simple... more In this work, we investigate extensions of the run-and-tumble particle model in 1D. In its simplest version, particle drift is limited to two velocities v = ±v 0 and the model is exactly solvable. Extension of the model to three drifts v = 0, ±v 0 yields the exact solution but the complexity of the expressions indicates that analytical treatment of higher-state models under the same procedure is impractical. Consequently, we modify our goal and consider a generalized version of the model for an arbitrary distribution of states P(v). To analyze such a system, we reformulate the Fokker–Planck equation as a self-consistent relation. The self-consistent relation is then analyzed by means of Laplace transform techniques.
In this work, we consider a lattice-gas model of charge regulation with electrostatic interaction... more In this work, we consider a lattice-gas model of charge regulation with electrostatic interactions within the Debye-Hückel level of approximation. In addition to long-range electrostatic interactions, the model incorporates the nearest-neighbor interactions for representing non-electrostatic forces between adsorbed ions. The Frumkin-Fowler-Guggenheim isotherm obtained from the mean-field analysis accurately reproduces the simulation data points.
This article examines the suggestion made in Ref. [EPL, 115 (2016) 60001] that a solution to a pa... more 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.
This work considers an extension of the Kuramoto model with run-and-tumble dynamics-a type of sel... more This work considers an extension of the Kuramoto model with run-and-tumble dynamics-a type of selfpropelled 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(ω). Because the Kuramoto model undergoes phase transition, it offers a simple case study for investigating phase transition for a system with self-propelled particles.
Uploads
Papers by Derek Frydel