scholarly journals From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 8 ◽  
Author(s):  
François Gay-Balmaz ◽  
Hiroaki Yoshimura
Keyword(s):  
Nonequilibrium Thermodynamics ◽  
Classical Mechanics ◽  
Discrete Systems ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Irreversible Processes ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Continuum Systems

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite-dimensional case of discrete systems, as well as for the infinite-dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton’s principle, we show, with the help of thermodynamic systems with gradually increasing complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat, and mass transfer in both the adiabatically closed cases and open cases. On the continuum side, we illustrate our theory using the example of multicomponent Navier–Stokes–Fourier systems.

scholarly journals A free energy Lagrangian variational formulation of the Navier–Stokes–Fourier system

2019 ◽  
Vol 16 (supp01) ◽  
pp. 1940006 ◽  
Author(s):  
François Gay-Balmaz ◽  
Hiroaki Yoshimura
Keyword(s):  
Free Energy ◽  
Riemannian Manifolds ◽  
Variational Formulation ◽  
Nonequilibrium Thermodynamics ◽  
Discrete Systems ◽  
Navier Stokes ◽  
Infinite Dimensional ◽  
Variational Framework ◽  
Independent Variable ◽  
Continuum Systems

We present a variational formulation for the Navier–Stokes–Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite-dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in [F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics, Part II: Continuum systems, J. Geom. Phys. 111 (2017) 194–212] as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier–Stokes–Fourier system on Riemannian manifolds.

scholarly journals A new proof of Donoghue's interpolation theorem

2004 ◽  
Vol 2 (3) ◽  
pp. 253-265 ◽  
Author(s):  
Yacin Ameur
Keyword(s):  
Hilbert Space ◽  
Radon Measure ◽  
Interpolation Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Positive Radon Measure ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
New Interpretation ◽  
Necessary And Sufficient

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

scholarly journals Further Insights Into the Damping-Induced Self-Recovery Phenomenon

2019 ◽  
Vol 141 (4) ◽  
Author(s):  
Tejas Kotwal ◽  
Roshail Gerard ◽  
Ravi Banavar
Keyword(s):  
Large Amplitude ◽  
Mechanical Systems ◽  
Dimensional Case ◽  
Theoretical Interpretation ◽  
Underactuated Mechanical Systems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Bicycle Wheel

In a series of papers, Chang et al. proved and experimentally demonstrated a phenomenon in underactuated mechanical systems, that they termed “damping-induced self-recovery.” This paper further investigates a few features observed in these demonstrated experiments and provides additional theoretical interpretation for the same. In particular, we present a model for the infinite-dimensional fluid–stool–wheel system, that approximates its dynamics to that of the better understood finite dimensional case, and comment on the effect of the intervening fluid on the large amplitude oscillations observed in the bicycle wheel–stool experiment.

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

scholarly journals ENTANGLED MARKOV CHAINS GENERATED BY SYMMETRIC CHANNELS

2005 ◽  
Vol 08 (03) ◽  
pp. 497-504 ◽  
Author(s):  
TAKAYUKI MIYADERA
Keyword(s):  
Markov Chains ◽  
Entropy Density ◽  
Dimensional Case ◽  
Von Neumann Entropy ◽  
Quantum Random Walk ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Symmetric Channel ◽  
Finite Dimensional ◽  
Finite Dimensional Case

A notion of entangled Markov chain was introduced by Accardi and Fidaleo in the context of quantum random walk. They proved that, in the finite-dimensional case, the corresponding states have vanishing entropy density, but they did not prove that they are entangled. In this note this entropy result is extended to the infinite-dimensional case under the assumption of finite speed of hopping. Then the entanglement problem is discussed for spin-1/2, entangled Markov chains generated by a binary symmetric channel with hopping probability 1-q. The von Neumann entropy of these states, restricted on a sublattice is explicitly calculated and shown to be independent of the size of the sublattice. This is a new, purely quantum, phenomenon. Finally the entanglement property between the sublattices [Formula: see text] and [Formula: see text] is investigated using the PPT criterium. It turns out that, for q≠ 0, 1, ½ the states are non-separable, thus truly entangled, while for q = 0, 1, ½, they are separable.

scholarly journals Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions

2010 ◽  
Vol 53 (3) ◽  
pp. 466-474 ◽  
Author(s):  
Luis Dubarbie
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Dimensional Case ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Disjointness Preserving ◽  
Vector Valued

AbstractIn this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

scholarly journals Real parts of quasi-nilpotent operators

1979 ◽  
Vol 22 (3) ◽  
pp. 263-269 ◽  
Author(s):  
P. A. Fillmore ◽  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Dimensional Case ◽  
Nilpotent Operator ◽  
The Real ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Nilpotent Operators ◽  
Finite Dimensional Case ◽  
Adjoint Operators

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

Local affine selections of convex set-valued functions

2019 ◽  
Vol 26 (4) ◽  
pp. 621-624
Author(s):  
Szymon Wąsowicz
Keyword(s):  
Banach Spaces ◽  
Main Part ◽  
Convex Set ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case

Abstract It is well known that not every convex set-valued function admits an affine selection. One could ask whether there exists at least one local affine selection. The answer is positive in the finite-dimensional case. The main part of the note comprises two examples of the non-existence of local affine selections of convex set-valued functions defined on certain infinite-dimensional Banach spaces.

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