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2001 (1)
1994 (1)
1Author    Günter RadonsRequires cookie*
 Title    Mechanisms for Phase Transitions in the Multifractal Analysis of Invariant Densities of Correlated Random Maps  
 Abstract    Multifractal properties of the invariant densities of correlated random maps are analyzed. It is proven that within the thermodynamical formalism phase transitions for finite correlations may be due to transients. For systems with infinite correlations we show analytically that phase transitions can occur as a consequence of localization-delocalization transitions of relevant eigenfunctions. 
  Reference    Z. Naturforsch. 49a, 1219—1222 (1994); received October 12 1994 
  Published    1994 
  Keywords    Fractals, Dynamical systems, Phase transitions, Stochastic automata, Random walk 
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 TEI-XML for    default:Reihe_A/49/ZNA-1994-49a-1219.pdf 
 Identifier    ZNA-1994-49a-1219 
 Volume    49 
2Author    B. H. LavendaRequires cookie*
 Title    Is Relativistic Quantum Mechanics Compatible with Special Relativity?  
 Abstract    The transformation from a time-dependent random walk to quantum mechanics converts a modi­ fied Bessel function into an ordinary one together with a phase factor e,ir/2 for each time the electron flips both direction and handedness. Causality requires the argument to be greater than the order o f the Bessel function. Assuming equal probabilities for jumps ± 1 , the normalized modified Bessel function o f an imaginary argument is the solution o f the finite difference differential Schrödinger equation whereas the same function o f a real argument satisfies the diffusion equation. In the nonrelativistic limit, the stability condition o f the difference scheme contains the mass whereas in the ultrarelativistic limit only the velocity of light appears. Particle waves in the nonrelativistic limit become elastic waves in the ultrarelativistic limit with a phase shift in the frequency and wave number of 7r/2. The ordinary Bessel function satisfies a second order recurrence relation which is a finite difference differential wave equation, using non-nearest neighbors, whose solutions are the chirality components o f a free-particle in the zero fermion mass limit. Reintroducing the mass by a phase transformation transforms the wave equation into the Klein-Gordon equation but does not admit a solution in terms o f ordinary Bessel functions. However, a sign change of the mass term permits a solution in terms o f a modified Bessel function whose recurrence formulas produce all the results of special relativity. The Lorentz transformation maximizes the integral o f the modified Bessel function and determines the paths o f steepest descent in the classical limit. If the definitions of frequency and wave number in terms o f the phase were used in special relativity, the condition that the frame be inertial would equate the superluminal phase velocity with the particle velocity in violation o f causality. In order to get surfaces o f constant phase to move at the group velocity, an integrating factor is required which determines how the intensity decays in time. The phase correlation between neighboring sites in quantum mechanics is given by the phase factor for the electron to reverse its direction, whereas, in special relativity, it is given by the Doppler shift. 
  Reference    Z. Naturforsch. 56a, 347—365 (2001); received February 14 2001 
  Published    2001 
  Keywords    Random Walks, Quantum Mechanics, Special Relativity, Ordinary and Modified Bessel Functions 
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 TEI-XML for    default:Reihe_A/56/ZNA-2001-56a-0347.pdf 
 Identifier    ZNA-2001-56a-0347 
 Volume    56