27 research outputs found

    The Function of the Second Postulate in Special Relativity

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    Many authors noted that the principle of relativity, together with space-time symmetries, suffices to derive Lorentz-like coordinate transformations between inertial frames. These contain a free parameter, kk, (equal to c2c^{-2} in special relativity) which is usually claimed to be empirically determinable, so that special relativity does not need the postulate of constancy of the speed of light. I analyze this claim and find that all methods destined to measure kk fail without further assumptions, similar to the second postulate. Specifically, measuring kk requires a signal that travels identically in opposite directions (this is unrelated to the conventionality of synchronization, as the one-postulate program implicitly selects the standard synchronization convention). Positing such a property about light is logically weaker than Einstein's second postulate but suffices to recover special relativity in full

    Theory of continuum percolation II. Mean field theory

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    I use a previously introduced mapping between the continuum percolation model and the Potts fluid to derive a mean field theory of continuum percolation systems. This is done by introducing a new variational principle, the basis of which has to be taken, for now, as heuristic. The critical exponents obtained are β=1\beta= 1, γ=1\gamma= 1 and ν=0.5\nu = 0.5, which are identical with the mean field exponents of lattice percolation. The critical density in this approximation is \rho_c = 1/\ve where \ve = \int d \x \, p(\x) \{ \exp [- v(\x)/kT] - 1 \}. p(\x) is the binding probability of two particles separated by \x and v(\x) is their interaction potential.Comment: 25 pages, Late

    Exact solution of a one-dimensional continuum percolation model

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    I consider a one dimensional system of particles which interact through a hard core of diameter \si and can connect to each other if they are closer than a distance dd. The mean cluster size increases as a function of the density ρ\rho until it diverges at some critical density, the percolation threshold. This system can be mapped onto an off-lattice generalization of the Potts model which I have called the Potts fluid, and in this way, the mean cluster size, pair connectedness and percolation probability can be calculated exactly. The mean cluster size is S = 2 \exp[ \rho (d -\si)/(1 - \rho \si)] - 1 and diverges only at the close packing density \rho_{cp} = 1 / \si . This is confirmed by the behavior of the percolation probability. These results should help in judging the effectiveness of approximations or simulation methods before they are applied to higher dimensions.Comment: 21 pages, Late

    Theory of continuum percolation III. Low density expansion

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    We use a previously introduced mapping between the continuum percolation model and the Potts fluid (a system of interacting s-states spins which are free to move in the continuum) to derive the low density expansion of the pair connectedness and the mean cluster size. We prove that given an adequate identification of functions, the result is equivalent to the density expansion derived from a completely different point of view by Coniglio et al. [J. Phys A 10, 1123 (1977)] to describe physical clustering in a gas. We then apply our expansion to a system of hypercubes with a hard core interaction. The calculated critical density is within approximately 5% of the results of simulations, and is thus much more precise than previous theoretical results which were based on integral equations. We suggest that this is because integral equations smooth out overly the partition function (i.e., they describe predominantly its analytical part), while our method targets instead the part which describes the phase transition (i.e., the singular part).Comment: 42 pages, Revtex, includes 5 EncapsulatedPostscript figures, submitted to Phys Rev

    Theory of continuum percolation I. General formalism

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    The theoretical basis of continuum percolation has changed greatly since its beginning as little more than an analogy with lattice systems. Nevertheless, there is yet no comprehensive theory of this field. A basis for such a theory is provided here with the introduction of the Potts fluid, a system of interacting ss-state spins which are free to move in the continuum. In the s1s \to 1 limit, the Potts magnetization, susceptibility and correlation functions are directly related to the percolation probability, the mean cluster size and the pair-connectedness, respectively. Through the Hamiltonian formulation of the Potts fluid, the standard methods of statistical mechanics can therefore be used in the continuum percolation problem.Comment: 26 pages, Late

    Theory of Second and Higher Order Stochastic Processes

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    This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial example is x¨=R(t)\ddot x = R(t), where R(t)R(t) is not a Gaussian white noise). The stochastic process is discretized into nn time-steps, all possible realizations are summed up and the continuum limit is taken. This procedure often yields closed form formulas for the joint probability distributions. Completely worked out examples include all Gaussian random forces and a large class of Markovian (non-Gaussian) forces. This approach is also useful for deriving Fokker-Planck equations for the probability distribution functions. This is worked out for Gaussian noises and for the Markovian dichotomous noise.Comment: 35 pages, PlainTex, accepted for publication in Phys Rev. E
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