71 research outputs found
Planar tautologies hard for resolution.
We prove exponential lower bounds on the resolution proofs of some tautologies, based on rectangular grid graphs. More specifically, we show a 2/sup /spl Omega/(n)/ lower bound for any resolution proof of the mutilated chessboard problem on a 2n/spl times/2n chessboard as well as for the Tseitin tautology (G. Tseitin, 1968) based on the n/spl times/n rectangular grid graph. The former result answers a 35 year old conjecture by J. McCarthy (1964)
Tree resolution proofs of the weak pigeon-hole principle.
We prove that any optimal tree resolution proof of PHPn m is of size 2&thetas;(n log n), independently from m, even if it is infinity. So far, only a 2Ω(n) lower bound has been known in the general case. We also show that any, not necessarily optimal, regular tree resolution proof PHPn m is bounded by 2O(n log m). To the best of our knowledge, this is the first time the worst case proof complexity has been considered. Finally, we discuss possible connections of our result to Riis' (1999) complexity gap theorem for tree resolution
Fluctuation-Induced Interactions Between Ellipsoidal Particle and Planar Substrate Immersed in Critical Medium
In our study we investigate the behaviour of the net force (NF) emerging
between an ellipsoidal particle and a thick plate (slab), when the interaction
takes place in a near critical fluid medium with account for the omnipresent
van der Waals forces (vdWF). Here we consider the case of complete wetting of
the objects surfaces by the medium, due to strong adsorbing local surface
potentials, exerted by thin solid coating films. The influence of the bulk
inner regions of the particle and the slab on the constituents of the fluid
results in long-ranged competing dispersion potentials. As a consequence from
the critical fluctuations of the medium, the system experiences an additional
effective interaction, traditionally termed critical Casimir force (CCF). The
forces of interest are evaluated numerically from integral expressions obtained
utilizing general scaling arguments and mean-field type calculations in
combination with the so-called "surface integration approach" (SIA). Within the
scenario considered here, this technique is applicable if one has knowledge of
the forces between two parallel semi-infinite plates, confining in between some
fluctuating fluid medium characterized by its temperature and chemical
potential . It is demonstrated that for a suitable set of particle-fluid,
slab-fluid, and fluid-fluid coupling parameters the competition between the
effects due to the coatings and the core regions of the objects result, when
one changes or , in {\it sign change} of the NF acting between the
ellipsoid and the slab.Comment: 8 pages, 2 figues. arXiv admin note: text overlap with
arXiv:1702.0491
Critical Casimir Effect: Exact Results
If a material body is immersed into a fluctuating medium, its shape and the
properties of its constituents modify the fluctuations in the surrounding
medium. If in the same medium there is a second body, modifications of the
fluctuation due to the first one influence the modifications due to the second
one. This mutual influence results in a force between these bodies. If the
fluctuating medium consists of the confined electromagnetic field in vacuum,
one speaks of the quantum mechanical Casimir effect. In the case that the order
parameter of material fields fluctuates - such as differences of number
densities or concentrations - and that the corresponding fluctuations of the
order parameter are long-ranged, one speaks of the critical Casimir effect.
This holds, e.g., in the case of systems which undergo a second-order phase
transition and which are thermodynamically located near the corresponding
critical point, or for systems with a continuous symmetry exhibiting Goldstone
mode excitations. Here we review the currently available exact results
concerning the critical Casimir effect in systems encompassing the
one-dimensional Ising, XY, and Heisenberg models, the two-dimensional Ising
model, the Gaussian and the spherical models, as well as the mean field results
for the Ising and the XY model. Special attention is paid to the influence of
the boundary conditions on the behavior of the Casimir force.Comment: 218 pages, 67 figure
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 systems the
statistical mechanics underlying such forces can be shown to differ in the
constant magnetic canonical ensemble (CE) from those in the
widely-studied constant 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 of its order parameter.
Here we derive exact result for the partition function of the one-dimensional
Ising model of spins and fixed value using the transfer matrix method
(TMM); earlier results obtained via the TMM were limited to and 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
Non-universal size dependence of the free energy of confined systems near criticality
The singular part of the finite-size free energy density of the O(n)
symmetric field theory in the large-n limit is calculated at finite
cutoff for confined geometries of linear size L with periodic boundary
conditions in 2 < d < 4 dimensions. We find that a sharp cutoff
causes a non-universal leading size dependence
near which dominates the universal scaling term . This
implies a non-universal critical Casimir effect at and a leading
non-scaling term of the finite-size specific heat above .Comment: RevTex, 4 page
Casimir force in the rotor model with twisted boundary conditions
We investigate the three dimensional lattice XY model with nearest neighbor
interaction. The vector order parameter of this system lies on the vertices of
a cubic lattice, which is embedded in a system with a film geometry. The
orientations of the vectors are fixed at the two opposite sides of the film.
The angle between the vectors at the two boundaries is where . We make use of the mean field approximation to study the mean
length and orientation of the vector order parameter throughout the film---and
the Casimir force it generates---as a function of the temperature , the
angle , and the thickness of the system. Among the results of that
calculation are a Casimir force that depends in a continuous way on both the
parameter and the temperature and that can be attractive or repulsive.
In particular, by varying and/or one controls \underline{both} the
sign \underline{and} the magnitude of the Casimir force in a reversible way.
Furthermore, for the case , we discover an additional phase
transition occurring only in the finite system associated with the variation of
the orientations of the vectors.Comment: 14 pages, 9 figure
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 -dimensional mean spherical model confined to a film geometry with
finite thickness is reported. Along the finite direction different kinds of
boundary conditions are applied: periodic , antiperiodic and free
surfaces with Dirichlet , Neumann and a combination of Neumann and
Dirichlet 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 . It is found, for ,
that the singular part of the free energy has the required finite-size scaling
form at the bulk critical temperature only for and . For the
remaining boundary conditions the standard finite-size scaling hypothesis is
not valid. At , the critical amplitude of the singular part of the free
energy (related to the so called Casimir amplitude) is estimated. We obtain
, and
, implying a fluctuation--induced attraction between
the surfaces for and repulsion in the other two cases. For and
we find a logarithmic dependence on .Comment: Version published in J. Phys. A: Math. Theo
Universality of the thermodynamic Casimir effect
Recently a nonuniversal character of the leading spatial behavior of the
thermodynamic Casimir force has been reported [X. S. Chen and V. Dohm, Phys.
Rev. E {\bf 66}, 016102 (2002)]. We reconsider the arguments leading to this
observation and show that there is no such leading nonuniversal term in systems
with short-ranged interactions if one treats properly the effects generated by
a sharp momentum cutoff in the Fourier transform of the interaction potential.
We also conclude that lattice and continuum models then produce results in
mutual agreement independent of the cutoff scheme, contrary to the
aforementioned report. All results are consistent with the {\em universal}
character of the Casimir force in systems with short-ranged interactions. The
effects due to dispersion forces are discussed for systems with periodic or
realistic boundary conditions. In contrast to systems with short-ranged
interactions, for one observes leading finite-size contributions
governed by power laws in due to the subleading long-ranged character of
the interaction, where is the finite system size and is the
correlation length.Comment: 11 pages, revtex, to appear in Phys. Rev. E 68 (2003
The bulk correlation length and the range of thermodynamic Casimir forces at Bose-Einstein condensation
The relation between the bulk correlation length and the decay length of
thermodynamic Casimir forces is investigated microscopically in two
three-dimensional systems undergoing Bose-Einstein condensation: the perfect
Bose gas and the imperfect mean-field Bose gas. For each of these systems, both
lengths diverge upon approaching the corresponding condensation point from the
one-phase side, and are proportional to each other. We determine the
proportionality factors and discuss their dependence on the boundary
conditions. The values of the corresponding critical exponents for the decay
length and the correlation length are the same, equal to 1/2 for the perfect
gas, and 1 for the imperfect gas
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