824 research outputs found
On the (Boltzmann) Entropy of Nonequilibrium Systems
Boltzmann defined the entropy of a macroscopic system in a macrostate as
the of the volume of phase space (number of microstates) corresponding
to . This agrees with the thermodynamic entropy of Clausius when
specifies the locally conserved quantities of a system in local thermal
equilibrium (LTE). Here we discuss Boltzmann's entropy, involving an
appropriate choice of macro-variables, for systems not in LTE. We generalize
the formulas of Boltzmann for dilute gases and of Resibois for hard sphere
fluids and show that for macro-variables satisfying any deterministic
autonomous evolution equation arising from the microscopic dynamics the
corresponding Boltzmann entropy must satisfy an -theorem.Comment: 31 pages, in Tex, authors' e-mails: [email protected],
[email protected]
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
Parallel single cell analysis on an integrated microfluidic platform for cell trapping, lysis and analysis
We report here a novel and easily scalable microfluidic platform for the parallel analysis of hundreds of individual cells, with controlled single cell trapping, followed by their lysis and subsequent retrieval of the cellular content for on-chip analysis. The device consists of a main channel and an array of shallow side channels connected to the main channel via trapping structures. Cells are individually captured in dam structures by application of a negative pressure from an outlet reservoir, lyzed on site and the cellular content controllably extracted and transported in the individual side channels for on-chip analysis.\u
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
Combined Lab-on-a-Chip and microarray approach for biomolecular interaction sensing using surface plasmon resonance imaging
Surface plasmon resonance imaging (SPR) is a well-established label-free detection technique for real-time biomolecular interaction measurements. An integrated LOC sensing system with fluidic control for sample movement to specific locations on microarray surface in combination with SPR imaging is demonstrated by the measurements of human IgG and anti-IgG interactions from 24 patterned regions.\u
Geometric inequalities from phase space translations
We establish a quantum version of the classical isoperimetric inequality
relating the Fisher information and the entropy power of a quantum state. The
key tool is a Fisher information inequality for a state which results from a
certain convolution operation: the latter maps a classical probability
distribution on phase space and a quantum state to a quantum state. We show
that this inequality also gives rise to several related inequalities whose
counterparts are well-known in the classical setting: in particular, it implies
an entropy power inequality for the mentioned convolution operation as well as
the isoperimetric inequality, and establishes concavity of the entropy power
along trajectories of the quantum heat diffusion semigroup. As an application,
we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck
semigroup, and argue that it implies fast convergence towards the fixed point
for a large class of initial states.Comment: 37 pages; updated to match published versio
On the Markov sequence problem for Jacobi polynomials
We give a simple and entirely elementary proof of Gasper's theorem on the
Markov sequence problem for Jacobi polynomials. It is based on the spectral
analysis of an operator that arises in the study of a probabilistic model of
colliding molecules introduced by Marc Kac. In the process, we obtain some new
integral formulas for ratios of Jacobi polynomials that generalize Gasper's
product formula and a well known formula of Koornwinder.Comment: The second version contains additional material and references. In
particular, we discuss a product formula of Koornwider and Schwarz, and show
how it may be proved using the methods developed here
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
Complete characterization of convergence to equilibrium for an inelastic Kac model
Pulvirenti and Toscani introduced an equation which extends the Kac
caricature of a Maxwellian gas to inelastic particles. We show that the
probability distribution, solution of the relative Cauchy problem, converges
weakly to a probability distribution if and only if the symmetrized initial
distribution belongs to the standard domain of attraction of a symmetric stable
law, whose index is determined by the so-called degree of
inelasticity, , of the particles: . This result is
then used: (1) To state that the class of all stationary solutions coincides
with that of all symmetric stable laws with index . (2) To determine
the solution of a well-known stochastic functional equation in the absence of
extra-conditions usually adopted
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