calcul variation entropie

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Calculus of variations and its applications

This thesis is about the calculus of variations The calculus of variations is a eld of mathematics about solving optimization problems This is done by minimizing and maximizing functionals The methods of calculus of variations to solve optimization problems are very useful in mathematics physics and engineering

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Introduction to the Modern Calculus of Variations

These lecture notes written for the MA4G6 Calculus of Variations course at the University of Warwick intend to give a modern introduction to the Calculus of Variations I have tried to cover different aspects of the field and to explain how they fit into the “big picture” This is not an encyclopedic work; many important results are omitted and s

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The Calculus of Variations

The calculus of variations is a ﬁeld of mathematics concerned with minimizing (or maximizing) functionals (that is real-valued functions whose inputs are functions) The calculus of variations has a wide range of applications in physics engineering applied and pure mathematics and is intimately connected to partial diﬀerential equations

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The Calculusof Variations

calculus of variations has continued to occupy center stage witnessing major theoretical advances along with wide-ranging applications in physics engineering and all branches of mathematics Minimization problems that can be analyzed by the calculus of variations serve to char-

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Chapitre IV 22ème principe de la thermodynamique

4 Variation d’entropie d’un système pour les transformations chimiques Entropie absolue Contrairement à U et H dont on peut mesurer que les variations on peut déterminer l’entropie absolue d’un corps qui est une mesure de son désordre moléculaire Entropie molaire standard (J K-1 mol-1): Rq 1: S(gaz)> S(liq) >S(solid)

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Thermodynamique

Entropie : définition thermodynamique Un système à température T reçoit une quantité de chaleur δQ au cours d'une transformation infinitésimale La variation d'entropie est donnée par : 1er principe:dU= Q W ⇒ Mathématiquement : U= U(SV) : dU= ∂U ∂S V dS ∂U ∂V S dV 2 dU=TdS−PdV 1 Si on compare (1) et (2) :

What does V mean in calculus of variations?
In equation (3.3), v is called the variation in the function y. We sometimes write v = y. We use the term variational derivative for the gradient. This is often written as J. The term variational is used a lot of times. This is where the name calculus of variations comes from.
What are classical solutions to minimization problems in the calculus of variations?
Classical solutions to minimization problems in the calculus of variations are prescribed by boundary value problems involving certain types of differential equations, known as the associated Euler–Lagrange equations.
How to prove isoperimetric inequality using Lagrange multiplier method?
We give here a short proof of the isoperimetric inequality (3.16) using the Lagrange multiplier method in the calculus of variations. Let the curve C be parametrized by (x(t); y(t)) for t 2 [0; 1]. For notational simplicity we write x = x1 and y = x2 in this section.
How to solve calculus of variations problems in Rd?
To address calculus of variations problems in Rd, we need to extend the notion of weak convergence to Sobolev spaces. Definition 4.14 (Weak convergence in W 1;p). Let 1 p < 1 and um; u 2 W 1;p(U). for all i 2 f1; : : : ; ng. Note we are using um * u for weak convergence in Lp and W 1;p spaces. It will be clear from the context which is intended.

Preface

These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and s

Introduction

Physical modeling is a delicate business – trying to balance the validity of the resulting model, that is, its agreement with nature and its predictive capabilities, with the feasibility of its mathematical treatment is often highly non-straightforward. In this quest to formulate a useful picture of an aspect of the world, it turns out on surprisin

Ñ φ

Ñ and the natural assumption that (E ) φ E ( ), the divergence theorem, warwick.ac.uk

1.3 Stationary states in quantum mechanics

The non-relativistic evolution of a quantum mechanical system with N degrees of freedom in an electrical field is described completely through its wave function : N warwick.ac.uk

2 R Ω

ics theory, we also need to require that y: y( ) is a differentiable bijection and that it warwick.ac.uk

F Ñ

Ω represents the total elastic energy stored in the system. If the elastic energy can be written in this way as ∫ W( y(x)) dx, we call the material hyperelastic. In applications, W is warwick.ac.uk

Convexity

In the introduction we got a glimpse of the many applications of minimization problems in physics, technology, and economics. In this chapter we start to develop the abstract theory. Consider a minimization problem of the form 8 ∫ [u] := f (x u(x) u(x)) dx min warwick.ac.uk

Ω R j Ω

Here, and throughout the course (if not otherwise stated) we will make the standard assump-tion that d is a bounded Lipschitz domain, that is, is open, bounded, and has a warwick.ac.uk

M(1 + vp + Ap)

jj j j for some M 0. Below, we will investigate the solvability and regularity properties of this minimization problem; in particular, we will take a close look at the way in which convexity properties of f in its gradient (third) argument determine whether is lower semicontinuous, which F is the decisive attribute in the fundamental Direct Method

L 2 R f 2 F Lg

atively compact, that is, if [xj] for a sequence (xj) X and some , then warwick.ac.uk

a F

liminf [xj] = j F a Thus, [x ] = and x is the sought minimizer. warwick.ac.uk

X with xj ⇀ x (weak conver-gence X) it holds that

[x] liminf [xj] F j F Then, the minimization problem [x] min over all x X, F 2 has at least one solution. Before we move on to the more concrete theory, let us record one further remark: While it might appear as if “nature does not give us the topology” and it is up to mathematicians to “invent” a suitable one, it is remarkable that the topology th

2.2 Functionals with convex integrands

Returning to the minimization problem at the beginning of this chapter, we first consider the minimization problem for the simpler functional ∫ [u] := f (x u(x)) dx warwick.ac.uk

Lemma 2.4.

Let f : Ω N be a Carath ́eodory integrand, that is, warwick.ac.uk

2 Ω 2 R m

This coercivity also suggests the exponent p for the definition of the function spaces where we look for solutions. We further assume the upper growth bound warwick.ac.uk

Ñ k

Thus, applying Fatou’s Lemma, liminf ( [uj] j F ) kjΩj warwick.ac.uk

2.3 Integrands with u-dependence

If we try to extend the results in the previous section to more general functionals ∫ [u] := f (x u(x) u(x)) dx min warwick.ac.uk

+ Ap)

j j and the convexity property A f (x v A) is convex for all (x v) m. warwick.ac.uk

2 Ω R

While it would be possible to give an elementary proof of this theorem here, we post-pone the verification to the next chapter. There, using more advanced and elegant techniques (blow-ups and Young measures), we will in fact establish a much more general result and see that, when viewed from the right perspective, u-dependence does not pose any add

Ω R R R

a Carath ́eodory integrand that is continuously differentiable in v and A and satisfies the growth bounds warwick.ac.uk

f (x v A))j

vj Hence we use the notation with the Frobenius matrix vector product “:” (see the appendix) and the scalar product “ ”. The boundary condition u = g on is to be understood in the Ω sense of trace. 3This is a system of PDEs, but we simply speak of it as an “equation”. Proof. For all Cc ( ; m) and all h warwick.ac.uk

2 Ω 2 R 2 R

This proposition is not difficult to establish and just needs a bit of notation; we omit the proof. One main use of the Euler–Lagrange equation is to find concrete solutions of variational problems: warwick.ac.uk

2 Ω R

solution. m) C( ; m) satisfies this PDE for every x , then we call u a classical warwick.ac.uk

Ñ Ñy Ñ y

Ω for all ( ; m). Integration by parts (more precisely, the Gauss–Green Theorem) y Cc warwick.ac.uk

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