161 research outputs found
Phase Transitions in "Small" systems
Traditionally, phase transitions are defined in the thermodynamic limit only.
We discuss how phase transitions of first order (with phase separation and
surface tension), continuous transitions and (multi)-critical points can be
seen and classified for small systems. Boltzmann defines the entropy as the
logarithm of the area W(E,N)=e^S(E,N) of the surface in the mechanical N-body
phase space at total energy E. The topology of the curvature determinant D(E,N)
of S(E,N) allows the classification of phase transitions without taking the
thermodynamic limit. The first calculation of the entire entropy surface S(E,N)
for the diluted Potts model (ordinary (q=3)-Potts model plus vacancies) on a
50*50 square lattice is shown. The regions in {E,N} where D>0 correspond to
pure phases, ordered resp. disordered, and D<0 represent transitions of first
order with phase separation and ``surface tension''. These regions are bordered
by a line with D=0. A line of continuous transitions starts at the critical
point of the ordinary (q=3)-Potts model and runs down to a branching point P_m.
Along this line \nabla D vanishes in the direction of the eigenvector v_1 of D
with the largest eigen-value \lambda_1\approx 0. It characterizes a maximum of
the largest eigenvalue \lambda_1. This corresponds to a critical line where the
transition is continuous and the surface tension disappears. Here the
neighboring phases are indistinguishable. The region where two or more lines
with D=0 cross is the region of the (multi)-critical point. The micro-canonical
ensemble allows to put these phenomena entirely on the level of mechanics.Comment: 21 pages,Latex, 12 eps figure
Constrained flow around a magnetic obstacle
Many practical applications exploit an external local magnetic field --
magnetic obstacle -- as an essential part of their constructions. Recently, it
has been demonstrated that the flow of an electrically conducting fluid
influenced by an external field can show several kinds of recirculation. The
present paper reports a 3D numerical study whose some results are compared with
an experiment about such a flow in a rectangular duct.Comment: accepted to JFM, 26 pages, 14 figure
Core of the Magnetic Obstacle
Rich recirculation patterns have been recently discovered in the electrically
conducting flow subject to a local external magnetic termed "the magnetic
obstacle" [Phys. Rev. Lett. 98 (2007), 144504]. This paper continues the study
of magnetic obstacles and sheds new light on the core of the magnetic obstacle
that develops between magnetic poles when the intensity of the external field
is very large. A series of both 3D and 2D numerical simulations have been
carried out, through which it is shown that the core of the magnetic obstacle
is streamlined both by the upstream flow and by the induced cross stream
electric currents, like a foreign insulated insertion placed inside the
ordinary hydrodynamic flow. The closed streamlines of the mass flow resemble
contour lines of electric potential, while closed streamlines of the electric
current resemble contour lines of pressure. New recirculation patterns not
reported before are found in the series of 2D simulations. These are composed
of many (even number) vortices aligned along the spanwise line crossing the
magnetic gap. The intensities of these vortices are shown to vanish toward to
the center of the magnetic gap, confirming the general conclusion of 3D
simulations that the core of the magnetic obstacle is frozen. The implications
of these findings for the case of turbulent flow are discussed briefly.Comment: 14 pages, 9 figures, submitted to Journal of Turbulenc
Thermodynamics of rotating self-gravitating systems
We investigate the statistical equilibrium properties of a system of
classical particles interacting via Newtonian gravity, enclosed in a
three-dimensional spherical volume. Within a mean-field approximation, we
derive an equation for the density profiles maximizing the microcanonical
entropy and solve it numerically. At low angular momenta, i.e. for a slowly
rotating system, the well-known gravitational collapse ``transition'' is
recovered. At higher angular momenta, instead, rotational symmetry can
spontaneously break down giving rise to more complex equilibrium
configurations, such as double-clusters (``double stars''). We analyze the
thermodynamics of the system and the stability of the different equilibrium
configurations against rotational symmetry breaking, and provide the global
phase diagram.Comment: 12 pages, 9 figure
On the analogy between streamlined magnetic and solid obstacles
Analogies are elaborated in the qualitative description of two systems: the
magnetohydrodynamic (MHD) flow moving through a region where an external local
magnetic field (magnetic obstacle) is applied, and the ordinary hydrodynamic
flow around a solid obstacle. The former problem is of interest both
practically and theoretically, and the latter one is a classical problem being
well understood in ordinary hydrodynamics. The first analogy is the formation
in the MHD flow of an impenetrable region -- core of the magnetic obstacle --
as the interaction parameter , i.e. strength of the applied magnetic field,
increases significantly. The core of the magnetic obstacle is streamlined both
by the upstream flow and by the induced cross stream electric currents, like a
foreign insulated insertion placed inside the ordinary hydrodynamic flow. In
the core, closed streamlines of the mass flow resemble contour lines of
electric potential, while closed streamlines of the electric current resemble
contour lines of pressure. The second analogy is the breaking away of attached
vortices from the recirculation pattern produced by the magnetic obstacle when
the Reynolds number , i.e. velocity of the upstream flow, is larger than a
critical value. This breaking away of vortices from the magnetic obstacle is
similar to that occurring past a real solid obstacle. Depending on the inlet
and/or initial conditions, the observed vortex shedding can be either symmetric
or asymmetric.Comment: minor changes, accepted for PoF, 26 pages, 7 figure
The Cluster Expansion for the Self-Gravitating gas and the Thermodynamic Limit
We develop the cluster expansion and the Mayer expansion for the
self-gravitating thermal gas and prove the existence and stability of the
thermodynamic limit N, V to infty with N/V^{1/3} fixed. The essential
(dimensionless) variable is here eta = [G m^2 N]/[V^{1/3} T] (which is kept
fixed in the thermodynamic limit). We succeed in this way to obtain the
expansion of the grand canonical partition function in powers of the fugacity.
The corresponding cluster coefficients behave in the thermodynamic limit as
[eta/N]^{j-1} c_j where c_j are pure numbers. They are expressed as integrals
associated to tree cluster diagrams. A bilinear recurrence relation for the
coefficients c_j is obtained from the mean field equations in the Abel form. In
this way the large j behaviour of the c_j is calculated. This large j behaviour
provides the position of the nearest singularity which corresponds to the
critical point (collapse) of the self-gravitating gas in the grand canonical
ensemble. Finally, we discuss why other attempts to define a thermodynamic
limit for the self-gravitating gas fail.Comment: LaTex 12 pages, 1 figure .p
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