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Register a new userExcess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results
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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 $Ltoinfty$ 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 $xtoinfty$, with $2<sigma<4$ and $2<d+sigmaleq 6$, 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 $L^{d} {mathcal F}_C/k_BT approx Xi_0(L/xi_infty) + g_omega L^{-omega}Xiomega(L/xi_infty) + g_sigma L^{-omega_sigm a} Xi_sigma(L xi_infty)$. The contribution $propto g_sigma$ decays for $T eq T_{c,infty}$ 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 $g_omega$.
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A version of the Greens functions theory of the Van der Waals forces which can be conveniently used in the presence of spatial dispersion is presented. The theory is based on the fluctuation-dissipation theorem and is valid for interacting bodies, separated by vacuum. Objections against theories acounting for the spatial dispersion are discussed.
We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-sigma}$, where $sigma<d<2sigma$ and $0<sigmaleq2$. For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature $T_c$, as well as for $T>T_c$ and $T<T_c$. The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are investigated. While in the critical and under critical region the force is of the order of $L^{-d}$, for $T>T_c$ it decays as $L^{-d-sigma}$, where $L$ is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when $d-1<sigma$, as well as the case with $d-1>sigma$, when one observes a dimensional crossover from $d$ to a $d-1$ dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and $T<T_c$ is also derived. We have proven analytically that the excess free energy is always negative and monotonically increasing function of $T$ and $H$. For the Casimir force we have demonstrated that for any $sigma ge 1$ it is everywhere negative, i.e. an attraction between the surfaces bounding the system is to be observed. At $T=T_c$ the force is an increasing function of $T$ for $sigma>1$ and a decreasing one for $sigma<1$. For any $d$ and $sigma$ the minimum of the force at $T=T_c$ is always achieved at some $H e 0$.
We consider near-critical two-dimensional statistical systems at phase coexistence on the half plane with boundary conditions leading to the formation of a droplet separating coexisting phases. General low-energy properties of two-dimensional field theories are used in order to find exact analytic results for one- and two-point correlation functions of both the energy density and order parameter fields. The subleading finite-size corrections are also computed and interpreted within an exact probabilistic picture in which interfacial fluctuations are characterized by the probability density of a Brownian excursion. The analytical results are compared against high-precision Monte Carlo simulations we performed for the specific case of the Ising model. The numerical results are found to be in good agreement with the analytic results in absence of adjustable parameters. The explicit analysis of the closed-form expression for order parameter correlations reveals the long-ranged character of interfacial correlations and their confinement within the interfacial region. The analysis of correlations is then carried out in momentum space through the notion of interface structure factor, which we extend to the case of systems bounded by a flat wall. The presence of the wall and its associated entropic repulsion leads to a specific term in the interface structure factor which we identify.
We present results of a Monte Carlo study for the ferromagnetic Ising model with long range interactions in two dimensions. This model has been simulated for a large range of interaction parameter $sigma$ and for large sizes. We observe that the results close to the change of regime from intermediate to short range do not agree with the renormalization group predictions.
We present an approach for computing long-range van der Waals (vdW) interactions between complex molecular systems and arbitrarily shaped macroscopic bodies, melding atomistic treatments of electronic fluctuations based on density functional theory in the former, with continuum descriptions of strongly shape-dependent electromagnetic fields in the latter, thus capturing many-body and multiple scattering effects to all orders. Such a theory is especially important when considering vdW interactions at mesoscopic scales, i.e. between molecules and structured surfaces with features on the scale of molecular sizes, in which case the finite sizes, complex shapes, and resulting nonlocal electronic excitations of molecules are strongly influenced by electromagnetic retardation and wave effects that depend crucially on the shapes of surrounding macroscopic bodies. We show that these effects together can modify vdW interactions by orders of magnitude compared to previous treatments based on Casimir--Polder or non-retarded approximations, which are valid only at macroscopically large or atomic-scale separations, respectively.
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