348 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
Derivation and New Interpretation of the Lorentz Transformations and Einstein's Theorem of Velocity Addition
It is traditionally believed that the Lorentz transformations (LT) and
Einstein's theorem of velocity addition (ETVA), underlying special relativity,
cannot be obtained from non-relativistic (classical) mechanics. In the present
paper it is shown, however, that both the LT and the ETVA are derivable within
the framework of classical kinematics if the speeds of material points are
bounded above by a certain universal limit which can coincide with the
speed of light in a vacuum.Comment: 8 page
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
Maguejo-Smolin Transformation as a Consequence of a Specific Definition of Mass, Velocity, and the Upper limit on Energy
We consider an alternative approach to non-linear special-relativistic
theories. The point of departure is not -deformed algebra (or even
group-theoretical considerations) but rather 3 physical postulates defining
particle's velocity, mass, and the upper bound on its energy in terms of the
respective classical quantities. For a specific definition of particle's
velocity we obtain Magueijo-Smolin (MS) version of the double
special-relativistic theory. It is shown that this version follows from the
-Poincare algebra by the appropriate choice of on the shell mass, such
that it is always less or equal Planck's mass. The -deformed
Hamiltonian is found which invalidates some arguments about unphysical
predictions of the MS transformation.Comment: 12 pages,1 figure, adding some detailed calculation
Generalized (s-Parameterized) Weyl Transformation
A general canonical transformation of mechanical operators of position and
momentum is considered. It is shown that it automatically generates a parameter
s which leads to a generalized (or s-parameterized) Wigner function. This
allows one to derive a generalized (s-parameterized) Moyal brackets for any
dimensions. In the classical limit the s-parameterized Wigner averages of the
momentum and its square yield the respective classical values. Interestingly
enough,in the latter case the classical Hamilton-Jacobi equation emerges as a
consequence of such a transition only if there is a non-zero parameter s.Comment: LaTeX (amsmath, amsextra) 16 pages, appendix (fixing LaTex
idiosincrasies); fixing some minor typo
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
Special relativity as classical kinematics of a particle with the upper bound on its speed. Part II. The general Lorentz transforrmation and the generalized velocity composition theorem
The kinematics of a particle with the upper bound on the particle's speed (a
modification of classical kinematics where such a restriction is absent) has
been developed in [arXiv:1204.5740]. It was based solely on classical mechanics
without employing any concepts , associated with the time dilatation or/and
length contraction. It yielded the 1-D Lorentz transformation (LT), free of
inconsistencies (inherent in the canonical derivation and interpretations of
the LT). Here we apply the same approach to derive the LT for the 3-dimensional
motion of a particle and the attendant law of velocity composition. As a
result, the infinite set of four-parameter transformations is obtained. The
requirement of linearity of these transformations selects out of this set the
two-parameter subset . The values of the remaining two parameters ,dictated by
physics of the motion, is explicitly determined , yielding the canonical form
of the 3-dimensional LT. The generalized law of velocity composition and the
attendant invariant ( not postulated apriori) of the motion are derived, As in
the one-dimensional case, present derivation, as a whole, does not have any
need in introducing the concepts of the time dilatation and length contraction,
and is based on the classical concepts of time and space.Comment: 10 page
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
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