Casimir forces can appear between intrusions placed in different media driven by several fluctuation mechanisms, either in equilibrium or out of it. Herein, we develop a general formalism to obtain such forces from the dynamical equations of the fluctuating medium, the statistical properties of the driving noise, and the boundary conditions of the intrusions (which simulate the interaction between the intrusions and the medium). As a result, an explicit formula for the Casimir force over the intrusions is derived. This formalism contains the thermal Casimir effect as a particular limit and generalizes the study of the Casimir effect to such systems through their dynamical equations, with no appeal to their Hamiltonian, if any exists. In particular, we study the Casimir force between two infinite parallel plates with Dirichlet or Neumann boundary conditions, immersed in several media with finite correlation lengths (reaction-diffusion system, liquid crystals, and two coupled fields with non-Hermitian evolution equations). The driving Gaussian noises have vanishing or finite spatial or temporal correlation lengths; in the first case, equilibrium is reobtained and finite correlations produce nonequilibrium dynamics. The results obtained show that, generally, nonequilibrium dynamics leads to Casimir forces, whereas Casimir forces are obtained in equilibrium dynamics if the stress tensor is anisotropic.

Loose granular structures stabilized against gravity by an effective cohesive force are investigated on a microscopic basis using contact dynamics. We study the influence of the granular Bond number on the density profiles and the generation process of packings, generated by ballistic deposition under gravity. The internal compaction occurs discontinuously in small avalanches and we study their size distribution. We also develop a model explaining the final density profiles based on insight about the collapse of a packing under changes of the Bond number.

We study self-avoiding random walks in an environment where sites are excluded randomly, in two and three dimensions. For a single polymer chain, we study the statistics of the time averaged monomer density and show that these are well described by multifractal statistics. This is true even far from the percolation transition of the disordered medium. We investigate solutions of chains in a disordered environment and show that the statistics cease to be multifractal beyond the screening length of the solution.

Based on a relativistic Maxwell-fluid description, the existence of ultrarelativistic laser-induced periodic waves in an electron-ion plasma is investigated. Within a one-dimensional propagation geometry nonlinear coupling of the electromagnetic and electrostatic components occurs that makes the fourth-order problem nonintegrable. A Hamiltonian description is derived, and the manifolds of periodic solutions are studied by Poincaré section plots. The influence of ion motion is investigated in different intensity regimes. For ultrarelativistic laser intensities the phase-space structures change significantly compared to the weakly relativistic case. Ion motion becomes very important such that finally electron-ion plasmas in the far-ultrarelativistic regime behave similarly to electron-positron plasmas. The characteristic new types of periodic solutions of the system are identified and discussed.

Ion and neutral parameters are determined in the high electron density, magnetized, hydrogen plasma beam of an ITER divertor relevant plasma via measurements of the $n=2$ excited neutrals. Ion rotation velocity (up to 7 km/s) and temperature (2–$3\hspace{0.5em}\mathrm{eV}~T_e$) are obtained from analysis of $\mathrm{H}\alpha$ spectra measured close to the plasma source. The methodology for neutral density determination is explained whereby measurements in the linear plasma beam of Pilot-PSI are compared to modeling. Ground-state atomic densities are obtained via the production rate of $n=2$ and the optical thickness of the Lyman-$\alpha$ transition (escape factor $~$0.6) and yield an ionization degree $>$85% and dissociation degree in the residual gas of $~$4%. A 30% proportion of molecules with a rovibrational excitation of more than 2 eV is deduced from the production rate of $n=2$ atoms. This proportion increases by more than a factor of 4 for a doubling of the electron density in the transition to ITER divertor relevant electron densities, probably because of a large increase in the production and confinement of ground-state neutrals. Measurements are made using laser-induced fluorescence (LIF) and absorption, the suitability of which are evaluated as diagnostics for this plasma regime. Absorption is found to have a much better sensitivity than LIF, mainly owing to competition with background emission.

We discuss the front propagation in the $A+B\to 2A$ reaction under subdiffusion, which is described by continuous-time random walks with a heavy-tailed power-law waiting-time probability density function. Using a crossover argument, we discuss the two scaling regimes of the front propagation: an intermediate asymptotic regime given by the front solution of the corresponding continuous equation, and the final asymptotics, which is fluctuation dominated and therefore lays out of reach of the continuous scheme. We moreover show that the continuous reaction subdiffusion equation indeed possesses a front solution that decelerates and becomes narrow in the course of time. This continuous description breaks down for larger times when the front gets atomically sharp. We show that the velocity of such fronts decays in time faster than in the continuous regime.

We study one-dimensional diffusion of $N$ hard-core interacting Brownian particles driven by the space- and time-dependent external force. We give the exact solution of the $N$-particle Smoluchowski diffusion equation. In particular, we investigate the nonequilibrium energetics of two interacting particles under the time-periodic driving. The hard-core interaction induces entropic repulsion which differentiates the energetics of the two particles. We present exact time-asymptotic results which describe the mean energy, the accepted work and heat, and the entropy production of interacting particles, and we contrast these quantities against the corresponding ones for the noninteracting particles.

We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by $\Omega ⩽72/91$, $\omega =D\Omega ⩽3/2$, and ${\Delta }_1=\nu \omega ⩽2$, based upon Cardy’s result for the crossing probability on an annulus. The upper bounds are consistent with many previous measurements of site percolation on square and triangular lattices and new measurements for bond percolation, suggesting that they are exact. They also agree with exponents for hulls proposed recently by Aharony and Asikainen, based upon results of den Nijs. A corrections scaling form evidently applicable to site percolation is also found.

We investigate the statistical mechanics of chirality and biaxiality in liquid crystals through a variety of theoretical approaches, including Monte Carlo simulations, lattice mean-field theory, and Landau theory. All of these calculations show that there is an important interaction between cholesteric twist and biaxial order: The twist acts as a field on the biaxial order, and conversely, the biaxial order increases the twist, that is, reduces the pitch. We model the behavior of chiral biaxial liquid crystals as a function of temperature and discuss how the predictions can be tested in experiments.

We have presented a first-principle theory-based derivation of an exact expression for the solvent number-dependent electron-detachment energy of a solvated species in the thermodynamic limit. We also propose a generalized equation bridging the electron detachment energies for small and infinitely large clusters, thus providing a new route to calculate the ionization potential of a negatively charged ion from the electron-detachment energies of its stable hydrated clusters. Most importantly, it has the ability to predict the instability range of microhydrated anions. The calculated results for the ionization potential for a number of ions are found to be in good agreement with the available experimental results, and the predicted instability range for the doubly charged anions ${\mathrm{SO}}_4^{2-}$ and ${\mathrm{C}}_2$${\mathrm{O}}_4^{2-}$ is also consistent with experimental and ab initio results.

An alternative approach for the analysis of Langevin-type stochastic processes in the presence of strong measurement noise is presented. For the case of Gaussian distributed, exponentially correlated, measurement noise it is possible to extract the strength and the correlation time of the noise as well as polynomial approximations of the drift and diffusion functions of the underlying Langevin equation.

We present a method that enables the use of the forward-reverse (FR) method of Kosztin et al. on a broader range of problems in soft matter physics. Our method, which we call the oscillating forward-reverse (OFR) method, adds an oscillatory steering potential to the constant velocity steering potential of the FR method. This enables the calculation of the potential of mean force (PMF) in a single unidirectional oscillatory drift, rather than multiple drifts in both directions as required by the FR method. By following small forward perturbations with small reverse perturbations, the OFR method is able to generate a piecewise reverse path that follows the piecewise forward path much more closely than any practical set of paths used in the FR method. We calculate the PMF for four different systems: a dragged Brownian oscillator, a pair of atoms in a Lennard-Jones liquid, a Na${}^{+}$-Cl${}^{-}$ ion pair in an aqueous solution, and a deca-alanine molecule being stretched in an implicit solvent. In all cases, the PMF results are in good agreement with those published previously using various other methods, and, to our knowledge, we give for the first time PMFs calculated by nonequilibrium methods for the Lennard-Jones and Na${}^{+}$-Cl${}^{-}$ systems.

We explore the properties of the low-temperature phase of the O($n$) loop model in two dimensions by means of transfer-matrix calculations and finite-size scaling. We determine the stability of this phase with respect to several kinds of perturbations, including cubic anisotropy, attraction between loop segments, double bonds, and crossing bonds. In line with Coulomb gas predictions, cubic anisotropy and crossing bonds are found to be relevant and introduce crossover to different types of behavior. Whereas perturbations in the form of loop-loop attractions and double bonds are irrelevant, sufficiently strong perturbations of these types induce a phase transition of the Ising type, at least in the cases investigated. This Ising transition leaves the underlying universal low-temperature O($n$) behavior unaffected.

I investigate the problem of finding a statistical description of a complex many-body system whose invariant measure cannot be constructed stemming from classical thermodynamics ensembles. By taking solitons as a reference system and by employing a general formalism based on the Ablowitz-Kaup-Newell-Segur scheme, I demonstrate how to build an invariant measure and, within a one-dimensional phase space, how to develop a suitable thermodynamics. A detailed example is provided with a universal model of wave propagation, with reference to a transparent potential sustaining gray solitons. The system shows a rich thermodynamic scenario, with a free-energy landscape supporting phase transitions and controllable emergent properties. I finally discuss the origin of such behavior, trying to identify common denominators in the area of complex dynamics.

Recent advances in single-molecule chemistry have led to designs for artificial multipedal walkers that follow tracks of chemicals. We investigate the motion of a class of walkers, called molecular spiders, which consist of a rigid chemically inert body and several flexible enzymatic legs. The legs can reversibly bind to chemical substrates on a surface and through their enzymatic action convert them to products. The legs can also reversibly bind to products, but at a different rate. Antal and Krapivsky have proposed a model for molecular spider motion over regular one-dimensional lattices [T. Antal and P. L. Krapivsky, Phys. Rev. E 76, 021121 (2007).]. In the model the legs hop from site to site under constraints imposed by connection to a common body. The first time a leg visits a site, the site is an uncleaved substrate, and the leg hops from this site only once it has cleaved it into a product. This cleavage happens at a rate $r<1$, slower than dissociation from a product site, $r=1$. The effect of cleavage is to slow down the hopping rate for legs that visit a site for the first time. Along with the constraints imposed on the legs, this leads to an effective bias in the direction of unvisited sites that decreases the average time needed to visit $n$ sites. The overall motion, however, remains diffusive in the long time limit. We have reformulated the Antal-Krapivsky model as a continuous-time Markov process and simulated many traces of this process using kinetic Monte Carlo techniques. Our simulations show a previously unpredicted transient behavior wherein spiders with small $r$ values move superdiffusively over significant distances and times. We explain this transient period of superdiffusive behavior by describing the spider process as switching between two metastates: a diffusive state $D$ wherein the spider moves in an unbiased manner over previously visited sites, and a boundary state $B$ wherein the spider is on the boundary between regions of visited and unvisited sites and experiences a bias in the direction of unvisited sites. We show that while the spider remains in the $B$ state it moves ballistically in the direction of unvisited sites, and while the spider is in the $D$ state it moves diffusively. The relative amount of time the spider spends in the two states determines how superdiffusively the spider moves. We show that the $B$ state is Markovian, but the $D$ state is non-Markovian because the duration of a $D$ period depends on how many sites have been visited previously. As time passes the spider spends progressively more time in the $D$ state (moving diffusively) and less time in the $B$ state (moving ballistically). This explains both the transient superdiffusive motion and the eventual decay to diffusive motion as $t\to \infty$.

Spin-echo experiments are often said to constitute an instant of antithermodynamic behavior in a concrete physical system that violates the second law of thermodynamics. We argue that a proper thermodynamic treatment of the effect should take into account the correlations between the spin and the translational degrees of freedom of the molecules. To this end, we construct an entropy functional using Boltzmann macrostates that incorporate both spin and translational degrees of freedom. With this definition there is nothing special in the thermodynamics of spin echoes: dephasing corresponds to Hamiltonian evolution and leaves the entropy unchanged; dissipation increases the entropy. In particular, there is no phase of entropy decrease in the echo. We also discuss the definition of macrostates from the underlying quantum theory and we show that the decay of net magnetization provides a faithful measure of entropy change.

Percolation thresholds and critical exponents for universal scaling laws are computed for microstructures that derive from phase-transformation processes in two dimensions. The computed percolation threshold for nucleation and growth processes, $p_c\approx 0.6612$, is similar to those obtained by random placement of disks and greater than that of spinodal decomposition, $p_c\approx 0.4987$. Three critical exponents for scaling behavior were computed and do not differ significantly from universal values. The time evolution of a characteristic microstructural length was also computed: For spinodal decomposition, this length grows according to a power law after a short incubation period; for nucleation and growth, there are several transitions in the nature of the growth law. We speculate that the transitions in nucleation and growth derive from competing effects of coalescence at short times and then subsequent coarsening. Short-range order is present, but different, for both classes of microstructural evolution.

We report calculations of the density of maximally random jamming of one-component and binary hard-disk fluids. The theoretical structure used provides a common framework for description of the hard-disk liquid-to-hexatic, the liquid-to-hexagonal crystal, and the liquid to maximally random jammed state transitions. Our analysis is based on locating a particular bifurcation of the solutions of the integral equation for the inhomogeneous single-particle density at the transition between different spatial structures. The bifurcation of solutions we study is initiated from the dense metastable fluid, and we associate it with the limit of stability of the fluid, which we identify with the transition from the metastable fluid to a maximally random jammed state. For the one-component hard-disk fluid the predicted packing fraction at which the metastable fluid to maximally random jammed state transition occurs is 0.84, in excellent agreement with the experimental value 0.84 ± 0.02. The corresponding analysis of the limit of stability of a binary hard-disk fluid with specified disk-diameter ratio and disk composition requires extra approximations in the representations of the direct correlation function, the equation of state, and the number of order parameters accounted for. Keeping only the order parameter identified with the largest peak in the structure factor of the highest-density regular lattice with the same disk- diameter ratio and disk composition as the binary fluid, the predicted density of maximally random jamming is found to be 0.84–0.87, depending on the equation of state used, and very weakly dependent on the ratio of disk diameters and the fluid composition, in agreement with both experimental data and computer simulation data.

The similarity between quantum mechanics and thermodynamics is discussed. It is found that if the Clausius equality is imposed on the Shannon entropy and the analog of the quantity of heat, then the value of the Shannon entropy comes to formally coincide with that of the von Neumann entropy of the canonical density matrix, and pure-state quantum mechanics apparently transmutes into quantum thermodynamics. The corresponding quantum Carnot cycle of a simple two-state model of a particle confined in a one-dimensional infinite potential well is studied, and its efficiency is shown to be identical to the classical one.

We study the fractal properties of the two-dimensional (2D) discrete scale-invariant (DSI) rough surfaces. The contour lines of these rough surfaces show clear DSI. In the appropriate limit the DSI surfaces converge to the scale-invariant rough surfaces. The fractal properties of the 2D DSI rough surfaces apart from possessing the discrete scale-invariance property follow the properties of the contour lines of the corresponding scale-invariant rough surfaces. We check this hypothesis by calculating numerous fractal exponents of the contour lines by using numerical calculations. Apart from calculating the known scaling exponents, some other new fractal exponents are also calculated.