Path integrals for stiff polymers applied to membrane physics

Path integrals for fractional Brownian motion and fractional Gaussian noise

Wiener's path integral plays a central role in the study of Brownian motion. Here we derive exact path-integral representations for the more general fractional Brownian motion (FBM) and for its time derivative process, fractional Gaussian noise (FGN). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot, and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the FBM and FGN, relies on the theory of singular integral equations, and overcomes some technical difficulties by representing the action functional for the FBM in terms of the FGN for the subdiffusive FBM and in terms of the derivative of the FGN for the super-diffusive FBM. We also extend the formalism to include external forcing. The exact and explicit path-integral representations make inroads in the study of the FBM and FGN.

Casimir-like interactions and surface anchoring duality in bookshelf geometry of smectic-A liquid crystals

We analyze the transverse intersubstrate Casimir-like force, arising as a result of thermal fluctuations of the liquid crystalline layers of a smectic-A film confined between two planar substrates in a bookshelf geometry, in which the equidistant smectic layers are placed perpendicular to the bounding surfaces. We discuss the variation of the interaction force as a function of the intersubstrate separation in the presence of surface anchoring to the substrates, showing that the force induced by confined fluctuations is attractive and depends on the penetration length as well as the layer spacing. The strongest effect occurs for tightly confined fluctuations, in which the surface anchoring energy is set to infinity, where the force per area scales linearly with the thermal energy and inversely with the fourth power of the intersubstrate separation. By reducing the strength of the surface anchoring energy, the force first becomes weaker in magnitude but then increases in magnitude as the surface anchoring strength is further reduced down to zero, in which case the force tends to that obtained in the limit of strong anchoring.

Fluctuation mediated interactions due to rigidity mismatch and their effect on miscibility of lipid mixtures in multicomponent membranes

We consider how membrane fluctuations can modify the miscibility of lipid mixtures, that is to say how the phase diagram of a boundary-constrained membrane is modified when the membrane is allowed to fluctuate freely in the case of zero surface tension. In order for fluctuations to have an effect, the different lipid types must have differing Gaussian rigidities. We show, somewhat paradoxically, that fluctuation-induced interactions can be treated approximately in a mean-field type theory. Our calculations predict that, depending on the difference in bending and Gaussian rigidity of the lipids, membrane fluctuations can either favor or disfavor mixing.

Main phase transition in lipid bilayers: Phase coexistence and line tension in a soft, solvent-free, coarse-grained model

We devise a soft, solvent-free, coarse-grained model for lipid bilayer membranes. The nonbonded interactions take the form of a weighted-density functional, which allows us to describe the thermodynamics of self-assembly and packing effects of the coarse-grained beads in terms of a density expansion of the equation of state and weighting functions that regularize the microscopic bead densities, respectively. Identifying the length and energy scales via the bilayer thickness and the thermal energy scale, k(B)T, the model qualitatively reproduces key characteristics (e.g., bending rigidity, area per molecule, and compressibility) of lipid membranes. We employ this model to study the main phase transition between the fluid and the gel phase of the bilayer membrane. We accurately locate the phase coexistence using free energy calculations and also obtain estimates for the bare and the thermodynamic line tension.

Effects of dielectric disorder on van der Waals interactions in slab geometries

We analyze the effects of disorder on the thermal Casimir interaction for the case of two semi-infinite planar slabs across an intervening homogeneous unstructured dielectric. The semi-infinite bounding layers are assumed to be composed of plane-parallel layers of random dielectric materials. We show that the effective thermal Casimir interaction at long distances is self-averaging and can be written in the same form as the one between nonrandom media but with the effective dielectric tensor of the corresponding random media. On the contrary, the behavior at short distances becomes random, and thus sample dependent, dominated by the local values of the dielectric constants proximal to each other across the central homogeneous unstructured dielectric layer. We extend these results to the regime of intermediate slab separations by using perturbation theory for weak disorder as well as by extensive numerical simulations for a number of systems where the dielectric function has a log-normal distribution. Numerical simulation completely corroborates all the main features of the disorder dependent thermal Casimir interaction deduced analytically.

Out-of-equilibrium behavior of Casimir-type fluctuation-induced forces for free classical fields

We present a general method to study the nonequilibrium behavior of Casimir-type fluctuation-induced forces for classical free scalar field theories. In particular, we analyze the temporal evolution of the force toward its equilibrium value when the field dynamics is given by a general class of overdamped stochastic dynamics (including the model A and model B classes). The steady-state force is also analyzed for systems which have nonequilibrium steady states, for instance, where they are driven by colored noise. The key to the method is that the out of equilibrium force is computed by specifying an energy of interaction between the field and the surfaces in the problem. In general, we find that there is a mapping of the dynamical problem onto a corresponding static one and in the case where the latter can be solved, the full dynamical behavior of the force can be extracted. The method is used to compute the nonequilibrium Casimir force induced between two parallel plates by a fluctuating field, in the cases of Dirichlet, Neumann, and mixed boundary conditions. Various other examples, such as the fluctuation-induced force between inclusions in fluctuating media, are discussed.

One-dimensional counterion gas between charged surfaces: exact results compared with weak- and strong-coupling analyses

We evaluate exactly the statistical integral for an inhomogeneous one-dimensional (1D) counterion-only Coulomb gas between two charged boundaries and from this compute the effective interaction, or disjoining pressure, between the bounding surfaces. Our exact results are compared to the limiting cases of weak and strong couplings which are the same for 1D and three-dimensional (3D) systems. For systems with a large number of counterions it is found that the weak-coupling (mean-field) approximation for the disjoining pressure works perfectly and that fluctuations around the mean-field in 1D are much smaller than in 3D. In the case of few counterions it works less well and strong-coupling approximation performs much better as it takes into account properly the discreteness of the counterion charges.

Thermal Casimir effect with soft boundary conditions

We consider the thermal Casimir effect in systems of parallel plates coupled to a massless free field theory via quadratic interaction terms which suppress (i) the field on the plates and (ii) the gradient of the field in the plane of the plates. These boundary interactions correspond to (i) the presence of an electrolyte in the plates and (ii) a uniform field of dipoles, in the plates, which are polarizable in the plane of the plates. These boundary interactions lead to Robin-type boundary conditions in the case where there is no field outside the two plates. In the appropriate limit, in both cases Dirichlet boundary conditions are obtained but we show that in case (i) the Dirichlet limit breaks down at short interplate distances and in (ii) it breaks down at large distances. The behavior of the two plate system is also seen to be highly dependent on whether the system is open or closed. In addition we analyze the Casimir force on a third plate placed between two outer plates. The force acting on the central plate is shown to be highly sensitive to whether or not the fluctuating scalar field is present in the region exterior to the two confining plates.