arXiv subjects

Eytan Katzav

Publications and source records attributed to Eytan Katzav.

At least 19 recordsLinked to original sources

arXiv2026

The distribution of eccentricities in random regular graphs

We derive a closed-form analytical expression for the distribution of eccentricities (DoE) in random regular graphs (RRGs) that consist of $N$ nodes of degree $c$. The DoE is given by the tail distribution $P(E > \ell) \simeq 1 - \exp \left[ - \exp \left( - \frac{ e^{b \ell} - μ}β \right) \right]$, where the distance $\ell$ takes integer values, $b = \ln (c-1)$ is the shape parameter, $β= \frac{c-2}{c} N$ is the scale parameter and $μ= \frac{c-2}{c} N \ln N$ is the location parameter. By providing the full distribution rather than a single characteristic length scale, we present a detailed view of the large-scale structure. In spite of the fact that the degrees of all the nodes are the same, their eccentricities exhibit non-trivial variations. We derive a closed-form expression for the mean eccentricity, which is given by $\langle E \rangle \simeq \frac{\ln N}{\ln (c-1)} + \frac{\ln \ln N}{\ln (c-1)} - \frac{ \ln c - \ln (c-2) }{ \ln (c-1) } + \frac{1}{2}$. We calculate the mode of the DoE, which exhibits a staircase profile as a function of the network size. Interestingly, the mode is given by $E_{\rm mode} ={\rm Round} \left( \langle E \rangle \right)$, where ${\rm Round}( x )$ is the nearest integer to $x$. We also calculate the variance ${\rm Var}(E)$ and show that it exhibits oscillations as a function of the network size $N$. The results presented in this paper may serve as benchmarks for algorithmic approaches to eccentricity calculations in large sparse networks. The eccentricities are important in practical applications such as broadcasting and global dissemination, where the network performance is determined by the longest delay times.

cond-mat.stat-mech↗

arXiv2026

First-passage processes in a deterministic one-dimensional cellular automaton model of traffic flow

We present analytical results for first-passage processes in a deterministic one-dimensional cellular automaton (CA) model of traffic flow. Starting at time $t=0$ from a random initial state with car density p, at every time step $t\ge 1$ each car moves one step to the right if the cell on its right is empty, and is stopped if it is occupied by another car. The model, which coincides with CA rule 184 in Wolfram's numbering scheme, exhibits a continuous dynamical phase transition at $p=1/2$, between a low-density free-flowing phase and a high-density congested phase. Using the framework of first-passage processes, we derive a closed-form expression for the distribution $P(T_{FS}=t)$ of first-stopping (FS) times, which is the probability that a randomly selected car will be stopped for the first time at time $t$. We also obtain a closed-form expression for the stopping probability $P_S(t)$, which is the probability that a randomly selected car will be stopped at time $t$. In the low-density phase of $0<p<1/2$, the probability $P_S(t)$ yields a closed-form expression for the distribution $P(T_{LS}=t)$ of last-stopping (LS) times, which is the probability that a randomly selected car will be stopped for the last time at time $t$, beyond which it will move freely indefinitely. In this regime, we analyze the relation between the LS time and the number of stopping events $N_S$ which take place up to that time. We present closed-form expressions for the joint distribution $P(T_{LS}=t,N_S=n)$, for the two conditional distributions that emanate from it and for the marginal distribution $P(N_S=n)$. These results provide insight on the time scales of congestion and relaxation in deterministic traffic flow from the point of view of individual cars. In a broader context, they provide insight on complex relaxation processes that involve many interacting particles, such as deterministic surface growth.

cond-mat.stat-mech↗

arXiv2026

Structure and dynamics in the low-density phase of a two-dimensional cellular automaton model of traffic flow

We analyze the structure and dynamics in the low-density phase of the deterministic two-dimensional cellular automaton model of traffic flow introduced in [O. Biham, A.A. Middleton and D. Levine, Phys. Rev. A 46, R6124 (1992)]. The model consists of horizontally-oriented (H) cars that move to the right and vertically-oriented (V) cars that move downward, on a square lattice of size $L$ with periodic boundary conditions. Starting from a random initial state of density $p$, which is equally divided between the H and V-cars, the model exhibits a phase transition at a critical density $p_c$. For $p p_c$ it evolves toward a fully-jammed state or to an intermediate state of congested traffic. In the FFP states, the H and V-cars segregate into homogeneous diagonal bands, in which they move freely without obstruction. To analyze the convergence toward the FFP states we introduce a configuration-space distance measure $D(t)=D_{\parallel}(t)+D_{\perp}(t)$ between the state of the system at time $t$ and the set of FFP states. The $D_{\parallel}(t)$ term accounts for the interactions between homotypic pairs of H (or V) cars, while $D_{\perp}(t)$ accounts for the interactions between heterotypic pairs of H and V-cars. We show that in the FFP states $D(t)=0$, while in all the other states $D(t)>0$. As the system evolves toward the FFP states, there is a separation of time scales, where $D_{\parallel}(t)$ decays very fast while $D_{\perp}(t)$ decays much more slowly. Moreover, the time dependence of $D_{\perp}(t)$ is well fitted by an exponentially truncated power-law decay of the form $D_{\perp}(t)\sim t^{-γ} \exp(-t/τ_{\perp})$, where $τ_{\perp}$ depends on $L$ and $p$. The power-law decay suggests avalanche-like dynamics with no characteristic scale, while the exponential cutoff is imposed by the finite lattice size.

nlin.CG↗

arXiv2025

Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks

We present analytical results for the distribution of first return (FR) times of non-backtracking random walks (NBWs) on undirected configuration model networks consisting of $N$ nodes with degree distribution $P(k)$. We focus on the case in which the network consists of a single connected component. Starting from a random initial node $i$ at time $t=0$, an NBW hops into a random neighbor of $i$ at time $t=1$ and at each subsequent step it continues to hop into a random neighbor of its current node, excluding the previous node. We calculate the tail distribution $P ( T_{\rm FR} > t )$ of first return times from a random initial node to itself. It is found that $P ( T_{\rm FR} > t )$ is given by a discrete Laplace transform of the degree distribution $P(k)$. This result exemplifies the relation between structural properties of a network, captured by the degree distribution, and properties of dynamical processes taking place on the network. Using the tail-sum formula, we calculate the mean first return time ${\mathbb E}[ T_{\rm FR} ]$. Surprisingly, ${\mathbb E}[ T_{\rm FR} ]$ coincides with the result obtained from Kac's lemma that applies to simple random walks (RWs). We also calculate the variance ${\rm Var}(T_{\rm FR})$, which accounts for the variability of first return times between different NBW trajectories. We apply this formalism to Erd{\H o}s-Rényi networks, random regular graphs and configuration model networks with exponential and power-law degree distributions and obtain closed-form expressions for $P( T_{\rm FR} > t )$ as well as its mean and variance. These results provide useful insight on the advantages of NBWs over simple RWs in network exploration, sampling and search processes.

cond-mat.stat-mech↗

arXiv2025

The effect of preferential node deletion on the structure of networks that evolve via preferential attachment

We present analytical results for the effect of preferential node deletion on the structure of networks that evolve via node addition and preferential attachment. To this end, we consider a preferential-attachment-preferential-deletion (PAPD) model, in which at each time step, with probability $P_{\rm add}$ there is a growth step where an isolated node is added to the network, followed by the addition of $m$ edges, where each edge connects a node selected uniformly at random to a node selected preferentially in proportion to its degree. Alternatively, with probability $P_{\rm del}=1-P_{\rm add}$ there is a contraction step, in which a preferentially selected node is deleted and its links are erased. The balance between the growth and contraction processes is captured by the growth/contraction rate $η=P_{\rm add}-P_{\rm del}$. For $0 < η\le 1$ the overall process is of network growth, while for $-1\leη<0$ the overall process is of network contraction. Using the master equation and the generating function formalism, we study the time-dependent degree distribution $P_t(k)$. It is found that for each value of $m>0$ there is a critical value $η_c(m)=-(m-2)/(m+2)$ such that for $η_c(m)<η\le1$ the degree distribution $P_t(k)$ converges towards a stationary distribution $P_{\rm st}(k)$. In the special case of pure growth, where $η=1$, the model is reduced to a preferential attachment growth model and $P_{\rm st}(k)$ exhibits a power-law tail, which is a characteristic of scale-free networks. In contrast, for $η_c(m)<η<1$ the distribution $P_{\rm st}(k)$ exhibits an exponential tail, which has a well-defined scale.This implies a phase transition at $η=1$, in contrast with the preferential-attachment-random-deletion (PARD) model [B. Budnick, O. Biham and E. Katzav, J. Stat. Mech. 013401 (2025)], in which the power-law tail remains intact as long as $η>0$.

physics.soc-ph↗

arXiv2024

Phase transition in evolving networks that combine preferential attachment and random node deletion

Analytical results are presented for the structure of networks that evolve via a preferential-attachment-random-deletion (PARD) model in the regime of overall network growth and in the regime of overall contraction. The phase transition between the two regimes is studied. At each time step a node addition and preferential attachment step takes place with probability $P_{\rm add}$, and a random node deletion step takes place with probability $P_{\rm del} = 1 - P_{\rm add}$. The balance between growth and contraction is captured by the parameter $η= P_{\rm add} - P_{\rm del}$, which in the regime of overall network growth satisfies $0 < η\le 1$ and in the regime of overall network contraction $-1 \le η< 0$. Using the master equation and computer simulations we show that for $-1 < η< 0$ the time-dependent degree distribution $P_t(k)$ converges towards a stationary form $P_{\rm st}(k)$ which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for $0 < η\le 1$. Thus, the PARD model has a phase transition at $η=0$, which separates between two structurally distinct phases. At the transition, for $η=0$, the degree distribution exhibits a stretched exponential tail. While the stationary degree distribution in the phase of overall growth represents an asymptotic state, in the phase of overall contraction $P_{\rm st}(k)$ represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes.

cond-mat.stat-mech↗

arXiv2024

The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin

We present analytical results for the joint probability distribution $P(T_{FR}=t,S=s)$ of first return (FR) times t and of the number of distinct sites s visited by a random walk (RW) on a one dimensional lattice before returning to the origin. The RW on a one dimensional lattice is recurrent, namely the probability to return to the origin is $P_{R}=1$. However the mean $\langle T_{FR}\rangle$ of the distribution $P(T_{FR}=t)$ of first return times diverges. Similarly, the mean $\langle S\rangle$ of the distribution $P(S=s)$ of the number of distinct sites visited before returning to the origin also diverges. The joint distribution $P(T_{FR}=t,S=s)$ provides a formulation that controls these divergences and accounts for the interplay between the kinetic and geometric properties of first return trajectories. We calculate the conditional distributions $P(T_{FR}=t|S=s)$ and $P(S=s|T_{FR}=t)$. We find that the conditional expectation value of first return times of trajectories that visit s distinct sites is ${\mathbb E}[T_{FR}|S=s]=\frac{2}{3}(s^2+s+1)$, and the variance is $Var(T_{FR}|S=s)=\frac{4}{45}(s-1)(s+2)(s^2+s-1)$. We also find that in the asymptotic limit, the conditional expectation value of the number of distinct sites visited by an RW that first returns to the origin at time $t=2n$ is ${\mathbb E}[S|T_{FR}=2n] \simeq \sqrt{πn}$, and the variance is $Var(S|T_{FR}=2n) \simeq π\left(\fracπ{3}-1\right)n$. These results go beyond the important recent results of Klinger et al. [{\it Phys. Rev. E} {\bf 105}, 034116 (2022)], who derived a closed form expression for the generating function of the joint distribution, but did not go further to extract an explicit expression for the joint distribution itself. The joint distribution provides useful insight on the efficiency of random search processes, in which the aim is to cover as many sites as possible in a given number of steps.

cond-mat.stat-mech↗

arXiv2024

Asymptotic Matching the Self-Consistent Expansion to Approximate the Modified Bessel Functions of the Second Kind

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem - too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, $K_α(x)$.

cond-mat.stat-mech↗

arXiv2024

Effects of self-avoidance on the packing of stiff rods on ellipsoids

Using a statistical-mechanics approach, we study the effects of geometry and self-avoidance on the ordering of slender filaments inside non-isotropic containers, considering cortical microtubules in plant cells, and packing of genetic material inside viral capsids as concrete examples. Within a mean-field approximation, we show analytically how the shape of the container, together with self-avoidance, affects the ordering of the stiff rods. We find that the strength of the self-avoiding interaction plays a significant role in the preferred packing orientation, leading to a first-order transition for oblate cells, where the preferred orientation changes from azimuthal, along the equator, to a polar one, when self-avoidance is strong enough. While for prolate spheroids the ground state is always a polar-like order, strong self-avoidance results with a deep meta-stable state along the equator. We compute the critical surface describing the transition between azimuthal and polar ordering in the three dimensional parameter space (persistence length, eccentricity, and self-avoidance) and show that the critical behavior of this system is in fact related to the butterfly catastrophe model. We calculate the pressure and shear stress applied by the filament on the surface, and the injection force needed to be applied on the filament in order to insert it into the volume. We compare these results to the pure mechanical study where self-avoidance is ignored, and discuss similarities and differences.

cond-mat.soft↗

arXiv2023

The distribution of shortest path lengths on trees of a given size in subcritical Erdos-Renyi networks

In the subcritical regime Erdős-Rényi (ER) networks consist of finite tree components, which are non-extensive in the network size. The distribution of shortest path lengths (DSPL) of subcritical ER networks was recently calculated using a topological expansion [E. Katzav, O. Biham and A.K. Hartmann, Phys. Rev. E 98, 012301 (2018)]. The DSPL, which accounts for the distance $\ell$ between any pair of nodes that reside on the same finite tree component, was found to follow a geometric distribution of the form $P(L=\ell | L < \infty) = (1-c) c^{\ell - 1}$, where $0 < c < 1$ is the mean degree of the network. This result includes the contributions of trees of all possible sizes and topologies. Here we calculate the distribution of shortest path lengths $P(L=\ell | S=s)$ between random pairs of nodes that reside on the same tree component of a given size $s$. It is found that $P(L=\ell | S=s) = \frac{\ell+1}{s^{\ell}} \frac{(s-2)!}{(s-\ell-1)!}$. Surprisingly, this distribution does not depend on the mean degree $c$ of the network from which the tree components were extracted. This is due to the fact that the ensemble of tree components of a given size $s$ in subcritical ER networks is sampled uniformly from the set of labeled trees of size $s$ and thus does not depend on $c$. The moments of the DSPL are also calculated. It is found that the mean distance between random pairs of nodes on tree components of size $s$ satisfies ${\mathbb E}[L|S=s] \sim \sqrt{s}$, unlike small-world networks in which the mean distance scales logarithmically with $s$.

cond-mat.stat-mech↗

arXiv2023

Thermally driven elastic membranes are quasi-linear across all scales

We study the static and dynamic structure of thermally fluctuating elastic thin sheets by investigating the overdamped dynamic Föppl-von Kármán equation, in which the Föppl-von Kármán equation from elasticity theory is driven by white noise. This nonlinear equation is governed by a single nondimensional coupling parameter $g$ whose large and small values correspond to weak and strong nonlinear coupling respectively. By analysing the weak coupling case with ordinary perturbation theory and the strong coupling case with a self-consistent methodology known as the self-consistent expansion, precise analytic predictions for the static and dynamic structure factors are obtained. The maximum frequency $n_{\max}$ supported by the system plays a role in determining which of three possible classes such sheets belong to: (1) when $g\gg1$, the system is mostly linear with roughness exponent $ζ=1$ and dynamic exponent $z=4$, (2) when $g\ll2/n_{\max}$, the system is extremely nonlinear with roughness exponent $ζ=1/2$ and dynamic exponent $z=3$, (3) between these regimes, an intermediate behaviour is obtained in which a crossover occurs such that the nonlinear behaviour is observed for small frequencies while the linear behaviour is observed for large frequencies. The large frequency linear tail is found to have a significant impact on the small frequency behaviour of the sheet. Back-of-the-envelope calculations suggest that ultra-thin materials such as graphene lie in this intermediate regime. Despite the existence of these three distinct behaviours, the decay rate of the dynamic structure factor is related to the static structure factor as if the system were completely linear. This quasi-linearity occurs regardless of the size of $g$ and at all length scales. Numerical simulations confirm the existence of the three classes of behaviour and the quasi-linearity of all classes.

cond-mat.soft↗

arXiv2023

Dynamics of Fluctuating Thin Sheets Under Random Forcing

We study the dynamic structure factor of fluctuating elastic thin sheets subject to conservative (athermal) random forcing. In Steinbock, Katzav & Boudaoud, Phys. Rev. Research 4, 033096 (2022), the static structure factor of such a sheet was studied. In this paper, we recap the model developed there and investigate its dynamic properties. Using the self-consistent expansion (SCE), the time dependent two-point function of the height profile is determined and found to decay exponentially in time. Despite strong nonlinear coupling, the decay rate of the dynamic structure factor is found to coincide with the effective coupling constant for the static properties which suggests that the model under investigation exhibits certain quasi-linear behaviour. Confirmation of these results by numerical simulations is also presented.

cond-mat.soft↗

arXiv2023

The distribution of the number of cycles in directed and undirected random 2-regular graphs

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of $N$ nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree $k=2$, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with $\ln N$, while the length of the longest cycle scales with $N$. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles $\langle S \rangle$ scales with $\ln N$. Here we present exact analytical results for the distribution $P_N(S=s)$ of the number of cycles $s$ in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large $N$ limit. The moments and cumulants of $P_N(S=s)$ are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of $N$ objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

cond-mat.stat-mech↗

arXiv2022

Analytical results for the distribution of first-passage times of random walks on random regular graphs

We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of $N$ nodes of degree $c \ge 3$. Starting from a random initial node at time $t=0$, at each time step $t \ge 1$ an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution $P( T_{\rm FP} = t )$ of first-passage times from a random initial node $i$ to a random target node $j$, where $j \ne i$. We distinguish between FP trajectories whose backbone follows the shortest path (SPATH) from the initial node $i$ to the target node $j$ and FP trajectories whose backbone does not follow the shortest path ($\lnot {\rm SPATH}$). More precisely, the SPATH trajectories from the initial node $i$ to the target node $j$ are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network. Moreover, the shortest path between $i$ and $j$ on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly when the length $\ell_{ij}$ of the shortest path between the initial node $i$ and the target node $j$ is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

cond-mat.stat-mech↗

arXiv2022

Kaluza-Klein Dimensional Reduction From Elasticity Theory of Crumpled Paper

During the last century, two independent theories using the concept of dimensional reduction have been developed independently. The first, known as Föppl-von Kàrmàn theory, uses Riemannian geometry and continuum mechanics to study the shaping of thin elastic structures which could become as complex as crumpled paper. The second one, known as Kaluza-Klein theory, uses Minkowskian geometry and general relativity to unify fundamental interactions and gravity under the same formalism. Here we draw a parallel between these two theories in an attempt to use concepts from elasticity theory of plates to recover the Einstein-Maxwell equations. We argue that Kaluza-Klein theory belongs to the same conceptual group of theories as three-dimensional elasticity, which upon dimensional reduction leads to the Föppl-von Kàrmàn theory of two-dimensional elastic plates. We exploit this analogy to develop an alternative Kaluza-Klein formalism in the framework of elasticity theory in which the gravitational and electromagnetic fields are respectively associated with stretching-like and bending-like deformations. We show that our approach of dimensional reduction allows us to retrieve the Lagrangian densities of both gravitational, electromagnetic and Dirac spinors fields as well as the Lagrangian densities of mass and charge sources.

gr-qc↗

arXiv2022

The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion

We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion). To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability $P_{add}$ and a random node deletion step takes place with probability $P_{del}=1-P_{add}$. The balance between the growth and contraction processes is captured by the parameter $η=P_{add}-P_{del}$. The case of pure network growth is described by $η=1$. In case that $0<η<1$ the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where $-1<η<0$, the overall process is of network contraction, while in the special case of $η=0$ the expected size of the network remains fixed, apart from fluctuations. Using the master equation we obtain a closed form expression for the time dependent degree distribution $P_t(k)$. The degree distribution $P_t(k)$ includes a term that depends on the initial degree distribution $P_0(k)$, which decays as time evolves, and an asymptotic distribution $P_{st}(k)$. In the case of pure network growth ($η=1$) the asymptotic distribution $P_{st}(k)$ follows an exponential distribution, while for $-1<η<1$ it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth ($0 < η< 1$) the degree distribution $P_t(k)$ eventually converges to $P_{st}(k)$. In the case of overall network contraction ($-1 < η< 0$) we identify two different regimes. For $-1/3 < η< 0$ the degree distribution $P_t(k)$ quickly converges towards $P_{st}(k)$. In contrast, for $-1 < η< -1/3$ the convergence of $P_t(k)$ is initially very slow and it gets closer to $P_{st}(k)$ only shortly before the network vanishes.

cond-mat.stat-mech↗

arXiv2022

Nonlinear extension of Kolosov-Muskhelishvili stress function formalism

The method of stress-function in elasticity theory is a powerful analytical tool with applications to a wide range of physical systems, including defective crystals, fluctuating membranes, and more. A complex coordinates formulation of stress function, known as Kolosov-Muskhelishvili formalism, enabled the analysis of elastic problems with singular domains, particularly cracks, forming the basis for fracture mechanics. A shortcoming of this method is its limitation to linear elasticity, which assumes Hookean energy and linear strain measure. Under finite loads, the linearized strain fails to describe the deformation field adequately, reflecting the onset of geometric nonlinearity. The latter is common in materials experiencing large rotations, such as regions close to the crack tip or elastic metamaterials. While a nonlinear stress function formalism exists, Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for nonlinear stress function. The new formalism allows us to port methods from complex analysis to nonlinear elasticity and to solve nonlinear problems in singular domains. Upon implementing the method to the crack problem, we discover that nonlinear solutions strongly depend on the applied remote loads, excluding a universal form of the solution close to the crack tip and questioning the validity of previous studies of nonlinear crack analysis.

cond-mat.mtrl-sci↗

arXiv2022

The Structure of Fluctuating Thin Sheets Under Random Forcing

We propose a mathematical model to describe the athermal fluctuations of thin sheets driven by the type of random driving that might be experienced prior to weak crumpling. The model is obtained by merging the Föppl-von Kármán equations from elasticity theory with techniques from out-of-equilibrium statistical physics to obtain a nonlinear strongly coupled $ϕ^{4}$-Langevin field equation with spatially varying kernel. With the aid of the self-consistent expansion (SCE), this equation is analytically solved for the structure factor of a fluctuating sheet. In contrast to previous research which has suggested that the structure factor follows an anomalous power-law, we find that the structure factor in fact obeys a logarithmically corrected rational function. Numerical simulations of our model confirm the accuracy of our analytical solution.

cond-mat.soft↗