1,005 research outputs found
Granular Impact Model as an Energy-Depth Relation
Velocity-squared drag forces are common in describing an object moving
through a granular material. The resulting force law is a nonlinear
differential equation, and closed-form solutions of the dynamics are typically
obtained by making simplifying assumptions. Here, we consider a generalized
version of such a force law which has been used in many studies of granular
impact. We show that recasting the force law into an equation for the kinetic
energy versus depth, K(z), yields a linear differential equation, and thus
general closed-form solutions for the velocity versus depth. This approach also
has several advantages in fitting such models to experimental data, which we
demonstrate by applying it to data from 2D impact experiments. We also present
new experimental results for this model, including shape and depth dependence
of the velocity-squared drag force
Continuous phase transitions with a convex dip in the microcanonical entropy
The appearance of a convex dip in the microcanonical entropy of finite
systems usually signals a first order transition. However, a convex dip also
shows up in some systems with a continuous transition as for example in the
Baxter-Wu model and in the four-state Potts model in two dimensions. We
demonstrate that the appearance of a convex dip in those cases can be traced
back to a finite-size effect. The properties of the dip are markedly different
from those associated with a first order transition and can be understood
within a microcanonical finite-size scaling theory for continuous phase
transitions. Results obtained from numerical simulations corroborate the
predictions of the scaling theory.Comment: 8 pages, 7 figures, to appear in Phys. Rev.
Statistical mechanics of non-hamiltonian systems: Traffic flow
Statistical mechanics of a small system of cars on a single-lane road is
developed. The system is not characterized by a Hamiltonian but by a
conditional probability of a velocity of a car for the given velocity and
distance of the car ahead. Distribution of car velocities for various densities
of a group of cars are derived as well as probabilities of density fluctuations
of the group for different velocities. For high braking abilities of cars
free-flow and congested phases are found. Platoons of cars are formed for
system of cars with inefficient brakes. A first order phase transition between
free-flow and congested phase is suggested.Comment: 12 pages, 6 figures, presented at TGF, Paris, 200
Finite-size behaviour of the microcanonical specific heat
For models which exhibit a continuous phase transition in the thermodynamic
limit a numerical study of small systems reveals a non-monotonic behaviour of
the microcanonical specific heat as a function of the system size. This is in
contrast to a treatment in the canonical ensemble where the maximum of the
specific heat increases monotonically with the size of the system. A
phenomenological theory is developed which permits to describe this peculiar
behaviour of the microcanonical specific heat and allows in principle the
determination of microcanonical critical exponents.Comment: 15 pages, 7 figures, submitted to J. Phys.
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