The "Real" Gibbs Paradox and a Composition-Based Resolution.

There is no documented evidence to suggest that J. W. Gibbs did not recognize the indistinguishable nature of states involving the permutation of identical particles or that he did not know how to justify on a priori grounds that the mixing entropy of two identical substances must be zero. However, there is documented evidence to suggest that Gibbs was puzzled by one of his theoretical findings, namely that the entropy change per particle would amount to kBln2 when equal amounts of any two different substances are mixed, no matter how similar these substances may be, and would drop straight to zero as soon as they become exactly identical. The present paper is concerned with this latter version of the Gibbs paradox and, to this end, develops a theory characterising real finite-size mixtures as realisations sampled from a probability distribution over a measurable attribute of the constituents of the substances. In this view, two substances are identical, relative to this measurable attribute, if they have the same underlying probability distribution. This implies that two identical mixtures do not need to have identical finite-size realisations of their compositions. By averaging over composition realisations, it is found that (1) fixed composition mixtures behave as homogeneous single-component substances and (2) in the limit of a large system size, the entropy of mixing per particle shows a continuous variation from kBln2 to 0, as two different substances are made more similar, thereby resolving the "real" Gibbs paradox.

On the Logic of a Prior Based Statistical Mechanics of Polydisperse Systems: The Case of Binary Mixtures.

Most undergraduate students who have followed a thermodynamics course would have been asked to evaluate the volume occupied by one mole of air under standard conditions of pressure and temperature. However, what is this task exactly referring to? If air is to be regarded as a mixture, under what circumstances can this mixture be considered as comprising only one component called "air" in classical statistical mechanics? Furthermore, following the paradigmatic Gibbs' mixing thought experiment, if one mixes air from a container with air from another container, all other things being equal, should there be a change in entropy? The present paper addresses these questions by developing a prior-based statistical mechanics framework to characterise binary mixtures' composition realisations and their effect on thermodynamic free energies and entropies. It is found that (a) there exist circumstances for which an ideal binary mixture is thermodynamically equivalent to a single component ideal gas and (b) even when mixing two substances identical in their underlying composition, entropy increase does occur for finite size systems. The nature of the contributions to this increase is then discussed.

Multicomponent flow on curved surfaces: A vielbein lattice Boltzmann approach.

We develop and implement a finite difference lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. The standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focusing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of the torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our finite difference lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.

On the critical Casimir interaction between anisotropic inclusions on a membrane.

Using a lattice model and a versatile thermodynamic integration scheme, we study the critical Casimir interactions between inclusions embedded in a two-dimensional critical binary mixtures. For single-domain inclusions we demonstrate that the interactions are very long range, and their magnitudes strongly depend on the affinity of the inclusions with the species in the binary mixtures, ranging from repulsive when two inclusions have opposing affinities to attractive when they have the same affinities. When one of the inclusions has no preference for either of the species, we find negligible critical Casimir interactions. For multiple-domain inclusions, mimicking the observations that membrane proteins often have several domains with varying affinities to the surrounding lipid species, the presence of domains with opposing affinities does not cancel the interactions altogether. Instead we can observe both attractive and repulsive interactions depending on their relative orientations. With increasing number of domains per inclusion, the range and magnitude of the effective interactions decrease in a similar fashion to those of electrostatic multipoles. Finally, clusters formed by multiple-domain inclusions can result in an effective affinity patterning due to the anisotropic character of the Casimir interactions between the building blocks.

Phase Separation on Bicontinuous Cubic Membranes: Symmetry Breaking, Reentrant, and Domain Faceting.

We study the phase separation of binary lipid mixtures that form bicontinuous cubic phases. The competition between the nonuniform Gaussian membrane curvature and line tension leads to a very rich phase diagram, where we observe symmetry breaking of the membrane morphologies and reentrant phenomena due to the formation of bridges between segregated domains. Upon increasing the line tension contribution, we also find faceting of lipid domains that we explain using a simple argument based on the symmetry of the underlying surface and topology.

Devising a protocol-related statistical mechanics framework for granular materials.

Devising a statistical mechanics framework for jammed granular materials is a challenging task as those systems do not share some important properties required to characterize them with statistical thermodynamics tools. In a recent paper [Asenjo et al. Phys. Rev. Lett. 112, 098002 (2014)], a new definition of a granular entropy, which puts the protocol used to generate the packings at its roots, has been proposed. Following up these results, it is shown that the protocol used in Asenjo et al. can be recast as a canonical ensemble with a particular value of the temperature. Signature of gaussianity for large system sizes strongly suggests an asymptotic equivalence with a corresponding microcanonical ensemble where jammed states with certain basin volumes are sampled uniformly. We argue that this microcanonical ensemble is not Edwards's microcanonical ensemble and generalize this argument to other protocols.

Numerical calculation of granular entropy.

We present numerical simulations that allow us to compute the number of ways in which N particles can pack into a given volume V. Our technique modifies the method of Xu, Frenkel, and Liu [Phys. Rev. Lett. 106, 245502 (2011)] and outperforms existing direct enumeration methods by more than 200 orders of magnitude. We use our approach to study the system size dependence of the number of distinct packings of a system of up to 128 polydisperse soft disks. We show that, even though granular particles are distinguishable, we have to include a factor 1=N! to ensure that the entropy does not change when exchanging particles between systems in the same macroscopic state. Our simulations provide strong evidence that the packing entropy, when properly defined, is extensive. As different packings are created with unequal probabilities, it is natural to express the packing entropy as S = − Σ(i)p(i) ln pi − lnN!, where pi denotes the probability to generate the ith packing. We can compute this quantity reliably and it is also extensive. The granular entropy thus (re)defined, while distinct from the one proposed by Edwards [J. Phys. Condens. Matter 2, SA63 (1990)], does have all the properties Edwards assumed.

Protein-DNA electrostatics: toward a new paradigm for protein sliding.

Gene expression and regulation rely on an apparently finely tuned set of reactions between some proteins and DNA. Such DNA-binding proteins have to find specific sequences on very long DNA molecules and they mostly do so in the absence of any active process. It has been rapidly recognized that, to achieve this task, these proteins should be efficient at both searching (i.e., sampling fast relevant parts of DNA) and finding (i.e., recognizing the specific site). A two-mode search and variants of it have been suggested since the 1970s to explain either a fast search or an efficient recognition. Combining these two properties at a phenomenological level is, however, more difficult as they appear to have antagonist roles. To overcome this difficulty, one may simply need to drop the dichotomic view inherent to the two-mode search and look more thoroughly at the set of interactions between DNA-binding proteins and a given DNA segment either specific or nonspecific. This chapter demonstrates that, in doing so in a very generic way, one may indeed find a potential reconciliation between a fast search and an efficient recognition. Although a lot remains to be done, this could be the time for a change of paradigm.

Probing ergodicity in granular matter.

When a granular system is tapped, its volume changes. Here, using a well-defined macroscopic protocol, we prepare an ensemble of granular systems and track the statistics of volume changes as a function of the number of taps. This is in contrast to previous studies, which have focused on single trajectories and assumed ergodicity. We devise a new method to assess the convergence properties of a sequence of ensemble volume histograms and introduce a reasonable approximate version of an invariant histogram. We then compare these invariant histograms with histograms generated by sampling a long trajectory for one system and observe nonergodicity, which we quantify. Finally, we use the overlapping histogram method to assess potential compatibility with Edwards' canonical assumption. Our histograms are incompatible with this assumption.

Interaction regimes for oppositely charged plates with multivalent counterions.

Within a mean-field treatment of the interaction between two oppositely charged plates in a salt-free solution, the distance at which a transition from an attractive to a repulsive regime appears can be computed analytically. The mean-field description, however, breaks down under strong Coulombic couplings, which can be achieved at room temperature with multivalent counterions and highly charged surfaces. Making use of the contact theorem and simple physical arguments, we propose explicit expressions for the equation of state in several situations at short distances. The possibility of Bjerrum pair formation is addressed and is shown to have profound consequences on the interactions. To complete the picture, we consider the large-distance limit, from which schematic phase diagram discriminating attractive from repulsive regions can be proposed.

Effective interaction between charged nanoparticles and DNA.

We investigate the effective interaction mediated by salt ions between charged nanoparticles (NPs) and DNA. DNA is modeled as an infinite cylinder with a constant surface charge in an implicit solvent. Monte Carlo simulations are used to compute the free energy of the system described in the framework of the primitive model of electrolytes, which accounts for excluded volumes of salt ions. A mean-field Poisson-Boltzmann theory also allows us to compute the free energy and provides us with explicit formulae for its main characteristics (position and depth of the minimum). We intend here to identify the physical parameters that have a major impact on the NP-DNA interaction, in an attempt to evaluate physico-chemical properties which could play a role in genotoxicity or, which could be exploited for therapeutic use. Thus, we investigate the influence on the effective interaction of: the shape of the nanoparticle, the magnitude of the nanoparticle charge and its distribution, the value of the pH of the solution, the magnitude of Van der Waals interactions depending on the nature of the constitutive material of the NP (metal vs. dielectric). We show that for positively charged concave NPs the effective interaction is repulsive at short distance, so that it presents a minimum at distance from the DNA. This short-range repulsion is specific to indented particles and is a robust property that holds for a large range of materials and charge densities.

Slits, plates, and Poisson-Boltzmann theory in a local formulation of nonlocal electrostatics.

Polar liquids like water carry a characteristic nanometric length scale, the correlation length of orientation polarizations. Continuum theories that can capture this feature commonly run under the name of "nonlocal" electrostatics since their dielectric response is characterized by a scale-dependent dielectric function ε(q), where q is the wave vector; the Poisson(-Boltzmann) equation then turns into an integro-differential equation. Recently, "local" formulations have been put forward for these theories and applied to water, solvated ions, and proteins. We review the local formalism and show how it can be applied to a structured liquid in slit and plate geometries, and solve the Poisson-Boltzmann theory for a charged plate in a structured solvent with counterions. Our results establish a coherent picture of the local version of nonlocal electrostatics and show its ease of use when compared to the original formulation.

Nonspecific DNA-protein interaction: why proteins can diffuse along DNA.

Recent single molecule experiments have reported that DNA binding proteins (DNA-BPs) can diffuse along DNA. This suggests that interactions between proteins and DNA play a role during the target search even far from their specific site on DNA. Here we show by means of Monte Carlo simulations and analytical calculations that there is a counterintuitive repulsion between the two oppositely charged macromolecules at a nanometer range. For the concave shape of DNA-BPs, and for realistic protein charge densities, we find that the DNA-protein interaction free energy has a minimum at a finite surface-to-surface separation, in which proteins can easily slide. When a protein encounters its target, the free energy barrier is completely counterbalanced by the H-bond interaction, thus enabling the sequence recognition.