109 research outputs found
Economics from a Physicist's point of view: Stochastic Dynamics of Supply and Demand in a Single Market. Part I
Proceeding from the concept of rational expectations, a new dynamic model of
supply and demand in a single market with one supplier, one buyer, and one kind
of commodity is developed. Unlike the cob-web dynamic theories with adaptive
expectations that are made up of deterministic difference equations, the new
model is cast in the form of stochastic differential equations. The
stochasticity is due to random disturbances ("input") to endogenous variables.
The disturbances are assumed to be stationary to the second order with zero
means and given covariance functions. Two particular versions of the model with
different endogenous variables are considered. The first version involves
supply, demand, and price. In the second version the stock of commodity is
added. Covariance functions and variances of the endogenous variables
("output") are obtained in terms of the spectral theory of stochastic
stationary processes. The impact of both deterministic parameters of the model
and the random input on the stochastic output is analyzed and new conditions of
chaotic instability are found. If these conditions are met, the endogenous
variables undergo unbounded chaotic oscillations. As a result, the market that
would be stable if undisturbed loses stability and collapses. This phenomenon
cannot be discovered even in principle in terms of any cobweb deterministic
model.Comment: 10 pages, LaTe
On an Elementary Derivation of the Hamilton-Jacobi Equation from the Second Law of Newton
It is shown that for a relativistic particle moving in an electromagnetic
field its equations of motion written in a form of the second law of Newton can
be reduced with the help of elementary operations to the Hamilton-Jacobi
equation. The derivation is based on a possibility of transforming the equation
of motion to a completely antisymmetric form. Moreover, by perturbing the
Hamilton-Jacobi equation we obtain the principle of least action.\
The analogous procedure is easily extended to a general relativistic motion
of a charged relativistic particle in an electromagnetic field. It sis also
shown that the special-relativistic Hamilton-Jacobi equation for a free
particle allows one to easily demonstrate the wave-particle duality inherent to
this equation and, in addition, to obtain the operators of the four-momentum
whose eigenvalues are the classical four-momentumComment: 12 pages,1 figure Abstract is modified, and a few substantial points
missed in the first version are adde
On Some Physical Aspects of Planck-Scale Relativity:A Simplified Approach
The kinematics of the two-scale relativity theory (new relativity) is
revisited using a simplified approach. This approach allows us not only to
derive the dispersion equation introduced earlier by Kowalski-Glikman, but to
find an additional dispersion relation, and, even more important, to provide a
physical basis for such relations. They are explained by the fact that in the
observer invariant two-scale relativity (new relativity) the Planck constant
does nor remain constant anymore, but depends on the universal length scale.
This leads to the correct relation between energy and frequency at any scale.Comment: 16 pages,1 figure,LaTe
Schroedinger revisited:How the time-dependent wave equation follows from the Hamilton-Jacobi equation
It is shown how using the classical Hamilton-Jacobi equation one can arrive
at the time-dependent wave equation. Although the former equation was
originally used by E.Schroedinger to get the wave equation, we propose a
different approach. In the first place, we do not use the principle of least
action and, in addition, we arrive at the time-dependent equation, while
Schroedinger (in his first seminal paper) used the least action principle and
obtained the stationary wave equation. The proposed approach works for any
classical Hamilton-Jacobi equation. In addition, by introducing information
loss into the Hamilton-Jacobi equation we derive in an elementary fashion the
wave equations (ranging from the Shroedinger to Klein-Gordon, to Dirac
equations). We also apply this technique to a relativistic particle in the
gravitational field and obtain the respective wave equation. All this supports
't Hooft's proposal about a possibility of arriving at quantum description from
a classical continuum in the presence of information loss.Comment: 19 pages; Some corrections to Introduction and Conclusio
One-Dimensional Motion of Bethe-Johnson Gas
A one dimensional motion of the Bethe-Johnson gas is studied in a context of
Landau's hydrodynamical model of a nucleus-nucleon collision. The expressions
for the entropy change, representing a generalization of the previously known
results, are found. It is shown that these expressions strongly depend on an
equation of state for the baryonic matter.Comment: 24 pages, 5 figure
A comment on the paper "Deformed Boost Transformations that saturate at the Planck Scale" by N.B.Bruno,G.Amelino-Camelia, and J.Kowalski-Glikman
An alternative (simplified) derivation of the dispersion relation and the
expressions for the momentum-energy 4-vector given initially in [1]
is provided. It has turned out that in a rather "pedestrian" manner one can
obtain in one stroke not only the above relations but also the correct
dispersion relation in space, consistent with the value of a
velocity of a massless particle. This is achieved by considering the standard
Lorentz algebra for -space. A non-uniqueness of the choice of the
time-derivative in the presence of the finite length scale is discussed. It is
shown that such non-uniqueness does not affect the dispersion relation in
-space. albeit results in different dispersion relations in
-space depending on the choice of the definition of the time derivative.Comment: 9 pages, LaTe
Energy Dissipation in Quantum Computers
A method is described for calculating the heat generated in a quantum
computer due to loss of quantum phase information. Amazingly enough, this heat
generation can take place at zero temperature. and may explain why it is
impossible to extract energy from vacuum fluctuations. Implications for optical
computers and quantum cosmology are also briefly discussed.Comment: 7 page
Fuzziness in Quantum Mechanics
It is shown that quantum mechanics can be regarded as what one might call a
"fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction
to the classical "crisp" logic. Therefore classical mechanics can be viewed as
a crisp limit of a "fuzzy" quantum mechanics. Based on these considerations it
is possible to arrive at the Schroedinger equation directly from the
Hamilton-Jacobi equation. The link between these equations is based on the fact
that a unique ("crisp") trajectory of a classical particle emerges out of a
continuum of possible paths collapsing to a single trajectory according to the
principle of least action. This can be interpreted as a consequence of an
assumption that a quantum "particle" "resides" in every path of the continuum
of paths which collapse to a single(unique) trajectory of an observed classical
motion. A wave function then is treated as a function describing a
deterministic entity having a fuzzy character. As a consequence of such an
interpretation, the complimentarity principle and wave-particle duality can be
abandoned in favor of a fuzzy deterministic microoobject.Comment: 20 pages, 2 figures, Late
Transition From Quantum To Classical Mechanics As Information Localization
Quantum parallelism implies a spread of information over the space in
contradistinction to the classical mechanical situation where the information
is "centered" on a fixed trajectory of a classical particle. This means that a
quantum state becomes specified by more indefinite data. The above spread
resembles, without being an exact analogy, a transfer of energy to smaller and
smaller scales observed in the hydrodynamical turbulence. There, in spite of
the presence of dissipation (in a form of kinematic viscosity), energy is still
conserved. The analogy with the information spread in classical to quantum
transition means that in this process the information is also conserved. To
illustrate that, we show (using as an example a specific case of a coherent
quantum oscillator) how the Shannon information density continuously changes in
the above transition . In a more general scheme of things, such an analogy
allows us to introduce a "dissipative" term (connected with the information
spread) in the Hamilton-Jacobi equation and arrive in an elementary fashion at
the equations of classical quantum mechanics (ranging from the Schr\"{o}dinger
to Klein-Gordon equations). We also show that the principle of least action in
quantum mechanics is actually the requirement for the energy to be bounded from
below.Comment: 30 pages, 8 figure
On Energy Expenditure per Unit of the Amount of Information
It is shown that for an equilibrium state of time-symmetric system of
non-relativistic strings the energy per unit of information transfer (storage,
processing) obeys the Bekenstein conjecture. The result is based on a theorem
due to A.Kholevo relating the physical entropy and the amount of information.
Interestingly, the energy in question is the difference between the ensemble
average of the energy and the Helmholtz free energy.Comment: 6 page
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