arXiv subjects

E. Trizac

Publications and source records attributed to E. Trizac.

At least 19 recordsLinked to original sources

arXiv2025

A strong-coupling effective-field theory for asymmetrically charged plates with counterions only

We are interested in rationalizing the phenomenon of like-charge attraction between charged bodies, such as a pair of colloids, in the strong coupling regime. The two colloids are modelled as uniformly charged parallel plates, neutralized by mobile counterions. In an earlier work [Palaia et al., J. Phys. Chem. B 126, 3143 (2022)], we developed an effective-field theory for symmetric plates, stemming from the ground-state description that holds at infinite couplings. Here, we generalize the approach to the asymmetric case, where the plates bear charges of the same sign, but of different values. In the symmetric situation, the mobile ions, which are localized in the vicinity of the two plates, share equally between both of them. Here, the sharing is non-trivial, depending both on the coupling parameter and the distance between the plates. We thus introduce a counterion occupation parameter, that is determined variationally to ensure minimum of the free energy. The resulting analytical results for the pressure as a function of the plate-plate distance $d$ agree well with our Monte Carlo data, in a large interval of strong and intermediate coupling constants $\Xi$. We show in particular that within this description, there exists a range of large distances at which the attractive pressure features a $1/d^2$ behavior.

cond-mat.soft↗

arXiv2024

Fate of Boltzmann's breathers: kinetic theory perspective

The dynamics of a system composed of elastic hard particles confined by an isotropic harmonic potential are studied. In the low-density limit, the Boltzmann equation provides an excellent description, and the system does not reach equilibrium except for highly specific initial conditions: it generically evolves towards and stays in a breathing mode. This state is periodic in time, with a Gaussian velocity distribution, an oscillating temperature and a density profile that oscillates as well. We characterize this breather in terms of initial conditions, and constants of the motion. For low but finite densities, the analysis requires to take into account the finite size of the particles. Under well-controlle approximations, a closed description is provided, which shows how equilibrium is reached at long times. The (weak) dissipation at work erodes the breather's amplitude, while concomitantly shifting its oscillation frequency. An excellent agreement is found between Molecular Dynamics simulation results and the theoretical predictions for the frequency shift. For the damping time, the agreement is not as accurate as for the frequency and the origin of the discrepancies is discussed.

cond-mat.stat-mech↗

arXiv2024

On the optimal relaxation rate for the Metropolis algorithm in one dimension

We study the relaxation of the Metropolis Monte Carlo algorithm corresponding to a single particle trapped in a one-dimensional confining potential, with even jump distributions that ensure that the dynamics verifies detailed balance. Previous work suggested that, for smooth jump distributions, the fastest relaxation rate is obtained as a result of the competition between diffusive and rejection-dominated dynamics. In this work, we show that a new regime comes into play for two-peaked jump distributions, where the relaxation dynamics is neither dominated by diffusion nor rejection: the eigenmodes adopt an oscillatory form, reminiscent of charge density waves (CDW) -- thus we term this new regime the CDW regime. Using a combination of numerical and analytical techniques, the parameter regions corresponding to diffusion, rejection, and CDW are characterised, as well as the transition lines between them -- i.e. a phase diagram is built. The optimal relaxation rate is located at the triple point of phase coexistence, where the transition lines (diffusive-rejection, diffusive-CDW, and CDW-rejection) intersect. Our theoretical framework is checked versus the numerical diagonalisation of the master equation. We also briefly discuss more sophisticated attempts at optimising the relaxation rate to equilibrium.

cond-mat.stat-mech↗

arXiv2024

Interplay between an absorbing phase transition and synchronization in a driven granular system

Absorbing phase transitions (APTs) are widespread in non-equilibrium systems, spanning condensed matter, epidemics, earthquakes, ecology, and chemical reactions. APTs feature an absorbing state in which the system becomes entrapped, along with a transition, either continuous or discontinuous, to an active state. Understanding which physical mechanisms determine the order of these transitions represents a challenging open problem in non-equilibrium statistical mechanics. Here, by numerical simulations and mean-field analysis, we show that a quasi-2d vibrofluidized granular system exhibits a novel form of APT. The absorbing phase is observed in the horizontal dynamics below a critical packing fraction, and can be continuous or discontinuous based on the emergent degree of synchronization in the vertical motion. Our results provide a direct representation of a feasible experimental scenario, showcasing a surprising interplay between dynamic phase transition and synchronization.

cond-mat.stat-mech↗

arXiv2023

Non-equilibrium attractor for non-linear stochastic dynamics

We study the dynamical behaviour of mesoscopic systems in contact with a thermal bath, described either via a non-linear Langevin equation at the trajectory level -- or the corresponding Fokker-Planck equation for the probability distribution function at the ensemble level. Our focus is put on one-dimensional -- or $d$-dimensional isotropic -- systems in confining potentials, with detailed balance -- fluctuation-dissipation thus holds, and the stationary probability distribution has the canonical form at the bath temperature. When quenching the bath temperature to low enough values, a far-from-equilibrium state emerges that rules the dynamics over a characteristic intermediate timescale. Such a long-lived state has a Dirac-delta probability distribution function and attracts all solutions over this intermediate timescale, in which the initial conditions are immaterial while the influence of the bath is still negligible. Numerical evidence and qualitative physical arguments suggest that the above picture extends to higher-dimensional systems, with anisotropy and interactions.

cond-mat.stat-mech↗

arXiv2020

Finite-time adiabatic processes: derivation and speed limit

Obtaining adiabatic processes that connect equilibrium states in a given time represents a challenge for mesoscopic systems. In this paper, we explicitly show how to build these finite-time adiabatic processes for an overdamped Brownian particle in an arbitrary potential, a system that is relevant both at the conceptual and the practical level. This is achieved by jointly engineering the time evolutions of the binding potential and the fluid temperature. Moreover, we prove that the second principle imposes a speed limit for such adiabatic transformations: there appears a minimum time to connect the initial and final states. This minimum time can be explicitly calculated for a general compression/decompression situation.

cond-mat.mes-hall↗

arXiv2019

Electric double layers with surface charge modulations: Novel exact Poisson-Boltzmann solutions

Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics. We provide novel exact analytical solutions to this non-linear mean-field approach, for the diffuse layer of ions in the vicinity of a planar or a cylindrical macroion. While previously known solution are for homogeneously charged objects, the cases worked out exhibit a modulated surface charge --or equivalently surface potential-- on the macroion (wall) surface. In addition to asymptotic features at large distances from the wall, attention is paid to the fate of the contact theorem, relating the contact density of ions to the local wall charge density. For salt-free systems (counterions only), we make use of results pertaining to the two-dimensional Liouville equation, supplemented by an inverse approach. When salt is present, we invoke the exact two-soliton solution to the 2D sinh-Gordon equation. This leads to inhomogeneous charge patterns, that are either localized or periodic in space. Without salt, the electrostatic signature of a charge pattern on the macroion fades exponentially with distance for a planar macroion, while it decays as an inverse power-law for a cylindrical macroion. With salt, our study is limited to the planar geometry, and reveals that pattern screening is exponential.

cond-mat.soft↗

arXiv2019

When random walkers help solving intriguing integrals

We revisit a family of integrals that delude intuition, and that recently appeared in mathematical literature in connection with computer algebra package verification. We show that the remarkable properties displayed by these integrals become transparent when formulated in the language of random walks. In turn, the random walk view naturally leads to a plethora of nontrivial generalizations, that are worked out. Related complex identities are also derived, without the need of explicit calculation. The crux of our treatment lies in a causality argument where a message that travels at finite speed signals the existence of a boundary.

cond-mat.stat-mech↗

arXiv2018

Optimal work in a harmonic trap with bounded stiffness

We apply Pontryagin's principle to drive rapidly a trapped overdamped Brownian particle in contact with a thermal bath between two equilibrium states corresponding to different trap stiffness $κ$. We work out the optimal time dependence $κ(t)$ by minimising the work performed on the particle under the non-holonomic constraint $0\leqκ\leqκ_{\max}$, an experimentally relevant situation. Several important differences arise, as compared with the case of unbounded stiffness that has been analysed in the literature. First, two arbitrary equilibrium states may not always be connected. Second, depending on the operating time $t_{\text{f}}$ and the desired compression ratio $κ_{\text{f}}/κ_{\textı}$, different types of solutions emerge. Finally, the differences in the minimum value of the work brought about by the bounds may become quite large, which may have a relevant impact on the optimisation of heat engines.

cond-mat.stat-mech↗

arXiv2018

Shortcut to stationary regimes: a simple experimental demonstration

We introduce a reverse engineering approach to drive a RC circuit. This technique is implemented experimentally 1) to reach a stationary regime associated to a sinusoidal driving in very short amount of time, 2) to ensure a fast discharge of the capacitor, and 3) to guarantee a fast change of stationary regime associated to different driving frequencies. This work can be used as a simple experimental project dedicated to the computer control of a voltage source. Besides the specific example addressed here, the proposed method provides an original use of simple linear differential equation to control the dynamical quantities of a physical system, and has therefore a certain pedagogical value.

physics.class-ph↗

arXiv2018

Strong-coupling theory of counterions between symmetrically charged walls: from crystal to fluid phases

We study thermal equilibrium of classical pointlike counterions confined between symmetrically charged walls at distance $d$. At very large couplings when the counterion system is in its crystal phase, a harmonic expansion of particle deviations is made around the bilayer positions, with a free lattice parameter determined from a variational approach. For each of the two walls, the harmonic expansion implies an effective one-body potential at the root of all observables of interest in our Wigner Strong-Coupling expansion. Analytical results for the particle density profile and the pressure are in good agreement with numerical Monte Carlo data, for small as well as intermediate values of $d$ comparable with the Wigner lattice spacing. While the strong-coupling theory is extended to the fluid regime by using the concept of a correlation hole, the Wigner calculations appear trustworthy for all electrostatic couplings investigated. Our results significantly extend the range of accuracy of analytical equations of state for strongly interacting charged planar interfaces.

cond-mat.soft↗

arXiv2018

A correlation-hole approach to the electric double layer with counter-ions only

We study a classical system of identically charged counter-ions near a planar wall carrying a uniform surface charge density. The equilibrium statistical mechanics of the system depends on a single dimensionless coupling parameter. A new self-consistent theory of the correlation-hole type is proposed which leads to a modified Poisson-Boltzmann integral equation for the density profile, convenient for analytical progress and straightforward to solve numerically. The exact density profiles are recovered in the limits of weak and strong couplings. In contrast to previous theoretical attempts of the test-charge family, the density profiles fulfill the contact-value theorem at all values of the coupling constant, and exhibit the mean-field decay at asymptotically large distances from the wall, as expected. We furthermore show that the density corrections at large couplings exhibit the proper dependence on coupling parameter and distance to the charged wall. The numerical results for intermediate values of the coupling provide accurate density profiles which are in good agreement with those obtained by Monte-Carlo simulations. The crossover to mean-field behavior at large distance is studied in detail.

cond-mat.soft↗

arXiv2018

Like-charge attraction in a one-dimensional setting: the importance of being odd

From cement cohesion to DNA condensation, a proper statistical physics treatment of systems with long range forces is important for a number of applications in physics, chemistry, and biology. We compute here the effective force between fixed charged macromolecules, screened by oppositely charged mobile ions (counterions). We treat the problem in a one dimensional configuration, that allows for interesting discussion and derivation of exact results, remaining at a level of mathematical difficulty compatible with an undergraduate course. Emphasis is put on the counter-intuitive but fundamental phenomenon of like-charge attraction, that our treatment brings for the first time to the level of undergraduate teaching. The parity of the number of counterions is shown to play a prominent role, which sheds light on the binding mechanism at work when like-charge macromolecules do attract.

cond-mat.soft↗

arXiv2018

Planar screening by charge polydisperse counterions

We study how a neutralising cloud of counterions screens the electric field of a uniformly charged planar membrane plate, when the counterions are characterised by a distribution of charges (or valence), $n(q)$. We work out analytically the one-plate and two-plate cases, at the level of non-linear Poisson-Boltzmann theory. The (essentially asymptotic) predictions are successfully compared to numerical solutions of the full Poisson-Boltzmann theory, but also to Monte Carlo simulations. The counterions with smallest valence control the long-distance features of interactions, and may qualitatively change the results pertaining to the classic monodisperse case where all counterions have the same charge. Emphasis is put on continuous distributions $n(q)$, for which new power-laws can be evidenced, be it for the ionic density or the pressure, in the one- and two-plates situations respectively. We show that for discrete distributions, more relevant for experiments, these scaling laws persist in an intermediate but yet observable range. Furthermore, it appears that from a practical point of view, hallmarks of the continuous $n(q)$ behaviour is already featured by discrete mixtures with a relatively small number of constituents.

cond-mat.stat-mech↗

arXiv2016

Quasi-universality in mixed counterions systems

The screening of plate-plate interactions by counterions is an age-old problem. We revisit this classic question when counterions exhibit a distribution of charges. While it is expected that the long-distance regime of interactions is universal, the behaviour of the inter-plate pressure at smaller distances should a priori depend rather severely on the nature of the ionic mixture screening the plate charges. We show that is not the case, and that for comparable Coulombic couplings, different systems exhibit a quasi-universal equation of state.

cond-mat.stat-mech↗

arXiv2016

Bending stiff charged polymers: the electrostatic persistence length

Many charged polymers, including nucleic acids, are locally stiff. Their bending rigidity -- quantified by the persistence length --, depends crucially on Coulombic features, such as the ionic strength of the solution which offers a convenient experimental route for tuning the rigidity. While the classic Odijk-Skolnick-Fixman treatment fails for realistic parameter values, we derive a simple analytical formula for the electrostatic persistence length. It is shown to be in remarkable agreement with numerically obtained Poisson-Boltzmann theory results, thereby fully accounting for non-linearities, among which counter-ion condensation effects. Specified to double-stranded DNA, our work reveals that the widely used bare persistence length of 500\,Å is overestimated by some 20%

cond-mat.soft↗

arXiv2016

Mean-field beyond mean-field: the single particle view for moderately to strongly coupled charged fluids

In a counter-ion only charged fluid, Coulomb coupling is quantified by a single dimensionless parameter. Yet, the theoretical treatment of moderately to strongly coupled charged fluids is a difficult task, central to the understanding of a wealth of soft matter problems, including biological systems. We show that the corresponding coupling regime can be remarkably well described by a single particle treatment, which, at variance with previous works, takes due account of inter-ionic interactions. To this end, the prototypical problem of a planar charged dielectric interface is worked out. Testing our predictions against Monte Carlo simulation data reveals an excellent agreement.

cond-mat.soft↗

arXiv2016

The rich polymorphic behavior of Wigner bilayers

Self-assembly into target structures is an efficient material design strategy. Combining analytical calculations and computational techniques of evolutionary and Monte Carlo types, we report about a remarkable structural variability of Wigner bilayer ground states, when charges are confined between parallel charged plates. Changing the inter-layer separation, or the plate charge asymmetry, a cascade of ordered patterns emerges. At variance with the symmetric case phenomenology, the competition between commensurability features and charge neutralization leads to long range attraction, appearance of macroscopic charges, exotic phases, and non conventional phase transitions with distinct critical indices, offering the possibility of a subtle, but precise and convenient control over patterns.

cond-mat.soft↗