Casimir versus Helmholtz forces: Exact results

Recently, attention has turned to the issue of the ensemble dependence of fluctuation induced forces. As a noteworthy example, in $O(n)$ systems the statistical mechanics underlying such forces can be shown to differ in the constant $\vec{M}$ magnetic canonical ensemble (CE) from those in the widely-studied constant $\vec{h}$ grand canonical ensemble (GCE). Here, the counterpart of the Casimir force in the GCE is the \textit{Helmholtz} force in the CE. Given the difference between the two ensembles for finite systems, it is reasonable to anticipate that these forces will have, in general, different behavior for the same geometry and boundary conditions. Here we present some exact results for both the Casimir and the Helmholtz force in the case of the one-dimensional Ising model subject to periodic and antiperiodic boundary conditions and compare their behavior. We note that the Ising model has recently being solved in Phys.Rev. E {\bf 106} L042103(2022), using a combinatorial approach, for the case of fixed value $M$ of its order parameter. Here we derive exact result for the partition function of the one-dimensional Ising model of $N$ spins and fixed value $M$ using the transfer matrix method (TMM); earlier results obtained via the TMM were limited to $M=0$ and $N$ even. As a byproduct, we derive several specific integral representations of the hypergeometric function of Gauss. Using those results, we rigorously derive that the free energies of the CE and grand GCE are related to each other via Legendre transformation in the thermodynamic limit, and establish the leading finite-size corrections for the canonical case, which turn out to be much more pronounced than the corresponding ones in the case of the GCE.Comment: 33 pages, 7 figures. The derivations in Appendix C are simplifie

Finite-size effects in the spherical model of finite thickness

A detailed analysis of the finite-size effects on the bulk critical behaviour of the $d$-dimensional mean spherical model confined to a film geometry with finite thickness $L$ is reported. Along the finite direction different kinds of boundary conditions are applied: periodic $(p)$, antiperiodic $(a)$ and free surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for $(p)$ and $(a)$. For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At $d=3$, the critical amplitude of the singular part of the free energy (related to the so called Casimir amplitude) is estimated. We obtain $\Delta^{(p)}=-2\zeta(3)/(5\pi)=-0.153051...$, $\Delta^{(a)}=0.274543...$ and $\Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and $(N)$ we find a logarithmic dependence on $L$.Comment: Version published in J. Phys. A: Math. Theo

On the finite-size behavior of systems with asymptotically large critical shift

Exact results of the finite-size behavior of the susceptibility in three-dimensional mean spherical model films under Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions are presented. The corresponding scaling functions are explicitly derived and their asymptotics close to, above and below the bulk critical temperature $T_c$ are obtained. The results can be incorporated in the framework of the finite-size scaling theory where the exponent $\lambda$ characterizing the shift of the finite-size critical temperature with respect to $T_c$ is smaller than $1/\nu$, with $\nu$ being the critical exponent of the bulk correlation length.Comment: 24 pages, late

Charge and Density Fluctuations Lock Horns : Ionic Criticality with Power-Law Forces

How do charge and density fluctuations compete in ionic fluids near gas-liquid criticality when quantum mechanical effects play a role ? To gain some insight, long-range $\Phi^{{\mathcal{L}}}_{\pm \pm} / r^{d+\sigma}$ interactions (with $\sigma>0$), that encompass van der Waals forces (when $\sigma = d = 3$), have been incorporated in exactly soluble, $d$-dimensional 1:1 ionic spherical models with charges $\pm q_0$ and hard-core repulsions. In accord with previous work, when $d>\min \{\sigma, 2\}$ (and $q_0$ is not too large), the Coulomb interactions do not alter the ($q_0 = 0$) critical universality class that is characterized by density correlations at criticality decaying as $1/r^{d-2+\eta}$ with $\eta = \max \{0, 2-\sigma\}$. But screening is now algebraic, the charge-charge correlations decaying, in general, only as $1/r^{d+\sigma+4}$; thus $\sigma = 3$ faithfully mimics known \textit{non}critical $d=3$ quantal effects. But in the \textit{absence} of full ($+, -$) ion symmetry, density and charge fluctuations mix via a transparent mechanism: then the screening \textit{at criticality} is \textit{weaker} by a factor $r^{4-2\eta}$. Furthermore, the otherwise valid Stillinger-Lovett sum rule fails \textit{at} criticality whenever $\eta =0$ (as, e.g., when $\sigma>2$) although it remains valid if $\eta >0$ (as for $\sigma<2$ or in real $d \leq 3$ Ising-type systems).Comment: 8 pages, in press in J. Phys. A, Letters to the Edito

Excess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results

We consider systems confined to a $d$-dimensional slab of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to\infty$ and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as $b x^{-(d+\sigma)}$ as $x\to\infty$, with $2<\sigma<4$ and $2<d+\sigma\leq 6$, on the Casimir effect at and near the bulk critical temperature $T_{c,\infty}$, for $2<d<4$. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form L^{d} {\mathcal F}_C/k_BT \approx \Xi_0(L/\xi_\infty) + g_\omega L^{-\omega}\Xi\omega(L/\xi_\infty) + g_\sigma L^{-\omega_\sigm a} \Xi_\sigma(L \xi_\infty). The contribution $\propto g_\sigma$ decays for $T\neq T_{c,\infty}$ algebraically in $L$ rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and $L$. We derive exact results for spherical and Gaussian models which confirm these findings. In the case $d+\sigma =6$, which includes that of nonretarded van-der-Waals interactions in $d=3$ dimensions, the power laws of the corrections to scaling $\propto b$ of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy $\omega=\omega_\sigma=4-d$ that occurs for the spherical model when $d+\sigma=6$, in conjunction with the $b$ dependence of $g_\omega$.Comment: 28 RevTeX pages, 12 eps figures, submitted to PR

Casimir force in O(n) lattice models with a diffuse interface

On the example of the spherical model we study, as a function of the temperature $T$, the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry $\infty^{d-1}\times L$, where $2<d<4$ is the dimensionality of the system. We consider a system with nearest-neighbor anisotropic interaction constants $J_\parallel$ parallel to the film and $J_\perp$ across it. The model represents the $n\to\infty$ limit of O(n) models with antiperiodic boundary conditions applied across the finite dimension $L$ of the film. We observe that the Casimir amplitude $\Delta_{\rm Casimir}(d|J_\perp,J_\parallel)$ of the anisotropic $d$-dimensional system is related to that one of the isotropic system $\Delta_{\rm Casimir}(d)$ via $\Delta_{\rm Casimir}(d|J_\perp,J_\parallel)=(J_\perp/J_\parallel)^{(d-1)/2} \Delta_{\rm Casimir}(d)$. For $d=3$ we find the exact Casimir amplitude $\Delta_{\rm Casimir}= [ {\rm Cl}_2 (\pi/3)/3-\zeta (3)/(6 \pi)](J_\perp/J_\parallel)$, as well as the exact scaling functions of the Casimir force and of the helicity modulus $\Upsilon(T,L)$. We obtain that $\beta_c\Upsilon(T_c,L)=(2/\pi^{2}) [{\rm Cl}_2(\pi/3)/3+7\zeta(3)/(30\pi)] (J_\perp/J_\parallel)L^{-1}$, where $T_c$ is the critical temperature of the bulk system. We find that the effect of the helicity is thus strong that the Casimir force is repulsive in the whole temperature region.Comment: 15 pages, 3 figure

Simplicial complex entropy.

We propose an entropy function for simplicial complices. Its value gives the expected cost of the optimal encoding of sequences of vertices of the complex, when any two vertices belonging to the same simplex are indistinguishable. We focus on the computational properties of the entropy function, showing that it can be computed efficiently. Several examples over complices consisting of hundreds of simplices show that the proposed entropy function can be used in the analysis of large sequences of simplicial complices that often appear in computational topology applications

Critical dynamics in thin films

Critical dynamics in film geometry is analyzed within the field-theoretical approach. In particular we consider the case of purely relaxational dynamics (Model A) and Dirichlet boundary conditions, corresponding to the so-called ordinary surface universality class on both confining boundaries. The general scaling properties for the linear response and correlation functions and for dynamic Casimir forces are discussed. Within the Gaussian approximation we determine the analytic expressions for the associated universal scaling functions and study quantitatively in detail their qualitative features as well as their various limiting behaviors close to the bulk critical point. In addition we consider the effects of time-dependent fields on the fluctuation-induced dynamic Casimir force and determine analytically the corresponding universal scaling functions and their asymptotic behaviors for two specific instances of instantaneous perturbations. The universal aspects of nonlinear relaxation from an initially ordered state are also discussed emphasizing the different crossovers that occur during this evolution. The model considered is relevant to the critical dynamics of actual uniaxial ferromagnetic films with symmetry-preserving conditions at the confining surfaces and for Monte Carlo simulations of spin system with Glauber dynamics and free boundary conditions.Comment: 64 pages, 21 figure

Scaling and nonscaling finite-size effects in the Gaussian and the mean spherical model with free boundary conditions

We calculate finite-size effects of the Gaussian model in a L\times \tilde L^{d-1} box geometry with free boundary conditions in one direction and periodic boundary conditions in d-1 directions for 2<d<4. We also consider film geometry (\tilde L \to \infty). Finite-size scaling is found to be valid for d3 but logarithmic deviations from finite-size scaling are found for the free energy and energy density at the Gaussian upper borderline dimension d* =3. The logarithms are related to the vanishing critical exponent 1-\alpha-\nu=(d-3)/2 of the Gaussian surface energy density. The latter has a cusp-like singularity in d>3 dimensions. We show that these properties are the origin of nonscaling finite-size effects in the mean spherical model with free boundary conditions in d>=3 dimensions. At bulk T_c in d=3 dimensions we find an unexpected non-logarithmic violation of finite-size scaling for the susceptibility \chi \sim L^3 of the mean spherical model in film geometry whereas only a logarithmic deviation \chi\sim L^2 \ln L exists for box geometry. The result for film geometry is explained by the existence of the lower borderline dimension d_l = 3, as implied by the Mermin-Wagner theorem, that coincides with the Gaussian upper borderline dimension d*=3. For 3<d<4 we find a power-law violation of scaling \chi \sim L^{d-1} at bulk T_c for box geometry and a nonscaling temperature dependence \chi_{surface} \sim \xi^d of the surface susceptibility above T_c. For 2<d<3 dimensions we show the validity of universal finite-size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for T>=T_c.Comment: Submitted to Physical Review

Depth Lower Bounds in Stabbing Planes for Combinatorial Principles

Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph