1,133 research outputs found
Correspondence between geometrical and differential definitions of the sine and cosine functions and connection with kinematics
In classical physics, the familiar sine and cosine functions appear in two
forms: (1) geometrical, in the treatment of vectors such as forces and
velocities, and (2) differential, as solutions of oscillation and wave
equations. These two forms correspond to two different definitions of
trigonometric functions, one geometrical using right triangles and unit
circles, and the other employing differential equations. Although the two
definitions must be equivalent, this equivalence is not demonstrated in
textbooks. In this manuscript, the equivalence between the geometrical and the
differential definition is presented assuming no a priori knowledge of the
properties of sine and cosine functions. We start with the usual length
projections on the unit circle and use elementary geometry and elementary
calculus to arrive to harmonic differential equations. This more general and
abstract treatment not only reveals the equivalence of the two definitions but
also provides an instructive perspective on circular and harmonic motion as
studied in kinematics. This exercise can help develop an appreciation of
abstract thinking in physics.Comment: 6 pages including 1 figur
Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix
We consider the correlation functions of eigenvalues of a unidimensional
chain of large random hermitian matrices. An asymptotic expression of the
orthogonal polynomials allows to find new results for the correlations of
eigenvalues of different matrices of the chain. Eventually, we consider the
limit of the infinite chain of matrices, which can be interpreted as a time
dependent one-matrix model, and give the correlation functions of eigenvalues
at different times.Comment: Tex-Harvmac, 27 pages, submitted to Journ. Phys.
Langevin dynamics with dichotomous noise; direct simulation and applications
We consider the motion of a Brownian particle moving in a potential field and
driven by dichotomous noise with exponential correlation. Traditionally, the
analytic as well as the numerical treatments of the problem, in general, rely
on Fokker-Planck description. We present a method for direct numerical
simulation of dichotomous noise to solve the Langevin equation. The method is
applied to calculate nonequilibrium fluctuation induced current in a symmetric
periodic potential using asymmetric dichotomous noise and compared to
Fokker-Planck-Master equation based algorithm for a range of parameter values.
Our second application concerns the study of resonant activation over a
fluctuating barrier.Comment: Accepted in Journal of Statistical Mechanics: Theory and Experimen
A quantum-mechanical Maxwell's demon
A Maxwell's demon is a device that gets information and trades it in for
thermodynamic advantage, in apparent (but not actual) contradiction to the
second law of thermodynamics. Quantum-mechanical versions of Maxwell's demon
exhibit features that classical versions do not: in particular, a device that
gets information about a quantum system disturbs it in the process. In
addition, the information produced by quantum measurement acts as an additional
source of thermodynamic inefficiency. This paper investigates the properties of
quantum-mechanical Maxwell's demons, and proposes experimentally realizable
models of such devices.Comment: 13 pages, Te
Breakdown of universality in multi-cut matrix models
We solve the puzzle of the disagreement between orthogonal polynomials
methods and mean field calculations for random NxN matrices with a disconnected
eigenvalue support. We show that the difference does not stem from a Z2
symmetry breaking, but from the discreteness of the number of eigenvalues. This
leads to additional terms (quasiperiodic in N) which must be added to the naive
mean field expressions. Our result invalidates the existence of a smooth
topological large N expansion and some postulated universality properties of
correlators. We derive the large N expansion of the free energy for the general
2-cut case. From it we rederive by a direct and easy mean-field-like method the
2-point correlators and the asymptotic orthogonal polynomials. We extend our
results to any number of cuts and to non-real potentials.Comment: 35 pages, Latex (1 file) + 3 figures (3 .eps files), revised to take
into account a few reference
Stochastic Energetics of Quantum Transport
We examine the stochastic energetics of directed quantum transport due to
rectification of non-equilibrium thermal fluctuations. We calculate the quantum
efficiency of a ratchet device both in presence and absence of an external load
to characterize two quantifiers of efficiency. It has been shown that the
quantum current as well as efficiency in absence of load (Stokes efficiency) is
higher as compared to classical current and efficiency, respectively, at low
temperature. The conventional efficiency of the device in presence of load on
the other hand is higher for a classical system in contrast to its classical
counterpart. The maximum conventional efficiency being independent of the
nature of the bath and the potential remains the same for classical and quantum
systems.Comment: To be published in Phys. Rev.
Thermodynamics of adiabatic feedback control
We study adaptive control of classical ergodic Hamiltonian systems, where the
controlling parameter varies slowly in time and is influenced by system's state
(feedback). An effective adiabatic description is obtained for slow variables
of the system. A general limit on the feedback induced negative entropy
production is uncovered. It relates the quickest negentropy production to
fluctuations of the control Hamiltonian. The method deals efficiently with the
entropy-information trade off.Comment: 6 pages, 1 figur
Mesoscopic Transport Through Ballistic Cavities: A Random S-Matrix Theory Approach
We deduce the effects of quantum interference on the conductance of chaotic
cavities by using a statistical ansatz for the S matrix. Assuming that the
circular ensembles describe the S matrix of a chaotic cavity, we find that the
conductance fluctuation and weak-localization magnitudes are universal: they
are independent of the size and shape of the cavity if the number of incoming
modes, N, is large. The limit of small N is more relevant experimentally; here
we calculate the full distribution of the conductance and find striking
differences as N changes or a magnetic field is applied.Comment: 4 pages revtex 3.0 (2-column) plus 2 postscript figures (appended),
hub.pam.94.
Comparing families of dynamic causal models
Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This “best model” approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data
Theory of random matrices with strong level confinement: orthogonal polynomial approach
Strongly non-Gaussian ensembles of large random matrices possessing unitary
symmetry and logarithmic level repulsion are studied both in presence and
absence of hard edge in their energy spectra. Employing a theory of polynomials
orthogonal with respect to exponential weights we calculate with asymptotic
accuracy the two-point kernel over all distance scale, and show that in the
limit of large dimensions of random matrices the properly rescaled local
eigenvalue correlations are independent of level confinement while global
smoothed connected correlations depend on confinement potential only through
the endpoints of spectrum. We also obtain exact expressions for density of
levels, one- and two-point Green's functions, and prove that new universal
local relationship exists for suitably normalized and rescaled connected
two-point Green's function. Connection between structure of Szeg\"o function
entering strong polynomial asymptotics and mean-field equation is traced.Comment: 12 pages (latex), to appear in Physical Review
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