• M C Valsakumar

      Articles written in Pramana – Journal of Physics

    • On the quantisation of dissipative systems

      M C Valsakumar

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      Two methods of quantisation of dissipative systems are considered. It is shown that the phase space description of quantum mechanics permits computational simplification, when Kanai’s method is adopted. Since the Moyal Bracket is the same as the Poisson Bracket, for systems described by a most general explicitly time dependent quadratic Lagrangian, the phase space distribution can be obtained as the solution of the corresponding classical Langevin equations in canonical variables, irrespective of the statistical properties of the noise terms. This result remains true for arbitrary potentials too in an approximate sense. Also analysed are Dekker’s theory of quantisation, violation of uncertainty principle in that theory and the reason for the same.

    • Diffraction from a quasi-crystalline chain

      M C Valsakumar Vijay Kumar

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      We present a general formalism for diffraction from a one-dimensional quasicrystal with arbitrary length scales and sequences. The notion of sub-quasi-lattices is introduced and the effect of different basis on different sites is studied. The relevance of this work for the study of vibrational and electronic spectra of the chain is discussed.

    • Analysis of a nonlinear stochastic model of cooperative behaviour

      M C Valsakumar

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      A stochastic model of cooperative behaviour is analyzed with regard to its critical properties. A cumulant expansion to fourth order is used to truncate the infinite set of coupled evolution equations for the moments. Linear stability analysis is performed around all the permissible steady states. The method is shown to be incapable of reproducing the critical boundary and the nature of the phase transition. A linearization, which respects the symmetry of the potential, is proposed which reproduces all the basic features associated with the model. The dynamics predicted by this approximation is shown to agree well with the Monte-Carlo simulation of the nonlinear Langevin equation.

    • Diffusion controlled multiplicative process: typical versus average behaviour

      M C Valsakumar K P N Murthy

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      We investigate the dynamics of the number of particles diffusing in a multiplicative medium. We show that the typical behaviour of the growth process is different from the average. We develop a new formalism to study the average growth process and extend it to the calculation of higher moments and finally of the probability distribution. We show that the fluctuations of the growth process increase exponentially with time. We describe the interesting features of the distribution.

    • First passage time on a multifurcating hierarchical structure

      V Sridhar K P N Murthy M C Valsakumar

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      Asymptotic behaviour of the moments of the first passage time (FPT) on a one-dimensional lattice holding a multifurcating hierarchy of teeth is studied. There is a transition from ordinary to anomalous diffusion when the parameter controlling the relative sizes of the teeth, is varied with respect to the furcating number of the hierarchy. The scaling behaviour of the moments of FPT with the linear dimensions of the lattice segment indicates that in the anomalcus phase the probability density of the FPT is multifractal.

    • A myopic random walk on a finite chain

      S Revathi V Balakrishnan M C Valsakumar

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      We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,N) in discrete time, with ‘myopic boundary conditions’—a walker at 1 (orN) at timen moves to 2 (orN − 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of the walker oscillate about certain mean values asn → ∞, with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out.

    • Solid state effects during deuterium implantation into copper and titanium

      H K Sahu M C Valsakumar B Panigrahi K G M Nair K Krishan

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      Results of neutron counting experiments during deuterium implantation into titanium and copper are reported. Models for neutron yield have been developed by taking into account different solid state effects like energy degradation of incident ions, energy dependent d-d fusion cross section and diffusion of implanted deuterium possibly influenced by surface desorption and formation of metal deuterides. The asymptotic time dependence of the neutron yield during implantation has been compared with the experimental results. Using these results, solid state processes that might occur during deuterium implantation into these metals are inferred.

    • Algebraic structure of Nambu mechanics

      Debendranath Sahoo M C Valsakumar

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      Abstracting from Nambu’s work [1] on the generalization of Hamiltonian mechanics, we obtain the concept of a classical Nambu algebra of type I (CNA-I). Consistency requirement of time evolution of the trilinear Nambu bracket leads to a new five point identity (FPI). Incorporating the FPI into CNA-I, we obtain a classical Nambu algebra of type II (CNA-II). Nambu’s algorithm for generalized classical mechanics turns out to be compatible with CNA-II. Tensor product composition of two CNA-I’s results in another CNA-I whereas that of two CNA-II’s does not. This implies that interacting systems cannot be consistently treated in Nambu’s framework. It is shown that the recent generalization of Nambu mechanics based on an arbitrary Lie group (instead of the particular case of the rotation group as in the case of Nambu’s original algorithm) suggested by Biyalinicki-Birula and Morrison [2], is compatible with CNA-I but not with CNA-II. Relaxation of the commutative and associative observable product by making it nonassociative so as to arrive at the quantum counterpart meets with serious difficulties from the view point of tensor product composition property. Thus neither CNA-I nor CNA-II have quantum counterparts. Implications of our results are discussed with special reference to existing work on Nambu mechanics in the literature.

    • q-Deformation of algebraic structures of quantum and classical mechanics

      Debendranath Sahoo M C Valsakumar

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      There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.

    • Signature of chaos in power spectrum

      M C Valsakumar S V M Satyanarayana V Sridhar

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      We investigate the nature of the numerically computed power spectral densityP(f, N, τ) of a discrete (sampling time interval,τ) and finite (length,N) scalar time series extracted from a continuous time chaotic dynamical system. We highlight howP(f, N, τ) differs from the true power spectrum and from the power spectrum of a general stochastic process. Non-zeroτ leads to aliasing;P(f, N, τ) decays at high frequencies as [πτ/sinπτf]2, which is an aliased form of the 1/f2 decay. This power law tail seems to be a characteristic feature of all continuous time dynamical systems, chaotic or otherwise. Also the tail vanishes in the limit ofN → ∞, implying that the true power spectral density must be band width limited. In striking contrast the power spectrum of a stochastic process is dominated by a term independent of the length of the time series at all frequencies.

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