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NonEquilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
, 1999
"... . We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two differ ..."
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Cited by 53 (14 self)
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. We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a noncompact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...
Spectral Properties of Hypoelliptic Operators
 Commun. Math. Phys
, 2003
"... We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show t ..."
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Cited by 18 (0 self)
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We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show that an inequality of the form C(#u#0,# + + iy)u#0,0 ) holds for suitable # and C which are independent of R, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the FokkerPlanck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of H erau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+ iy # y c, # (0, 1], c R}.
Extensive Properties of the Complex GinzburgLandau Equation
, 1998
"... . We study the set of solutions of the complex GinzburgLandau equation in R d , d ! 3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube QL of side L. We cover this set by a (minimal) number NQ L (") of balls of radius ..."
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Cited by 6 (0 self)
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. We study the set of solutions of the complex GinzburgLandau equation in R d , d ! 3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube QL of side L. We cover this set by a (minimal) number NQ L (") of balls of radius " in L 1 (QL ). We show that the Kolmogorov " entropy per unit length, H " = lim L!1 L \Gammad log NQ L (") exists. In particular, we bound H " by O \Gamma log(1=") \Delta , which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H " ? O \Gamma log(1=") \Delta . 1. Introduction In the last few years, considerable effort has been made towards a better understanding of partial differential equations of parabolic type in infinite space. A typical equation is for example the complex GinzburgLandau equation (CGL) on R d : @ t A = (1 + iff)\DeltaA + A \Gamma (1 + ifi)AjAj 2 : (1:1) Such eq...
Asymptotics without Logarithmic Terms for Crack Problems
"... We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface M with boundary E . The same boundary conditions are prescribed on both sides of the surface M . The most important model behind this investigation is the crack problem in threedimensional linear ..."
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Cited by 4 (3 self)
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We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface M with boundary E . The same boundary conditions are prescribed on both sides of the surface M . The most important model behind this investigation is the crack problem in threedimensional linear elasticity (either isotropic or anisotropic): there the boundary conditions are Neumann, i.e. tractions are prescribed on both faces of the crack surface M . We prove that the singular functions appearing in the expansion of the solution along the crack edge E all have the #(#) in local polar coordinates (r, #) : the logarithmic shadow terms predicted by the general theory do not appear. Moreover, we obtain that, for a smooth right hand side, the jump of the displacement across the crack surface is the product of r with a smooth vector function on M . We present
TDISubspaces of C(R^d) and some Density Problems from Neural Networks
 J. Approx. Theory
, 1996
"... . In this paper we consider translation and dilation invariant subspaces of C(IR d ). We characterize all such subspaces and also identify those f 2 C(IR d ), the span of whose translates and dilates generate nontrivial subspaces. We apply these and related results to some mathematical models ..."
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. In this paper we consider translation and dilation invariant subspaces of C(IR d ). We characterize all such subspaces and also identify those f 2 C(IR d ), the span of whose translates and dilates generate nontrivial subspaces. We apply these and related results to some mathematical models in the theory of neural networks. x1. Introduction Some time ago Burkhard Lenze asked me the following question: For which functions oe 2 C(IR) is spanfoe / ae d Y i=1 (x i \Gamma b i ) ! : ae; b 1 ; : : : ; b d 2 IRg dense in C(IR d )? Here we consider C(IR d ) with the topology of uniform convergence on compact subsets. Lenze's interest in this question was prompted by a mathematical model from the theory of neural networks. More specifically, a type of multilayer feedforward network with a single hidden layer, see Lenze [11] and [12]. The consideration of this specific problem led us to the study of a more general question on translation and dilation invariant subspaces. In ...
Monotonicity and Symmetry of positive solutions to nonlinear elliptic equations: local Moving Planes and Unique Continuation
"... . We prove local properties of symmetry and monotonicity for nonnegative solutions of scalar field equations with nonlinearities which are not Lipschitz. Our main tools are a local Moving Planes method and a unique continuation argument which is connected with techniques used for proving the uniquen ..."
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. We prove local properties of symmetry and monotonicity for nonnegative solutions of scalar field equations with nonlinearities which are not Lipschitz. Our main tools are a local Moving Planes method and a unique continuation argument which is connected with techniques used for proving the uniqueness of radially symmetric solutions. 1 1 Introduction We consider the semilinear elliptic partial differential equation \Deltau+f (u) = 0, x 2 D ae IR N and its ground states, which are nonnegative solutions vanishing at the boundary. This equation, often called the nonlinear scalar field equation, plays an important role in various domains of Physics, Chemistry and Population Dynamics, and it is fundamental from the mathematical point of view. Regarding ground states of scalar field equations there are two strongly related natural questions: symmetry and uniqueness. Coffman in [4] proved that \Deltau \Gamma u + u 3 = 0, x 2 IR N has a unique radially symmetric ground state. Later ...