Cronograma de Actividades Diagrama de Gantt

To conclude the section on stochastic descriptions, we review some convenient tools for the manipulation of probability distributions. It is not our intention to be entirely rigorous, we refer to the literature [76] for a full mathematical treatment of these subjects.

The Radon-Nikod´ym derivative

Suppose we are given a probability densityρ(x) andµ(x) onR.

Definition 1.4.2 (Absolutely continuous)

We say that the density ρis absolutely continuous with respect to the density

µ, writtenρ_≪µ, if

We can ask if it is possible to relate expectation values under one probabil- ity distribution with the other. This is precisely the content of the Radon- Nikod´ym theorem14

Theorem 1.4.1 (Radon-Nikod´ym)

If ρ_≪µ, then there exists a measurable functionf :R→R

+ _{such that}

ρ(A) = Z

A

f dµ.

We write the Radon-Nikod´ym derivative f as

f = dρ dµ.

We will often work with measures ρwhich are absolutely continuous with re- spect to the Lebesgue measure “dx”. In this case, we can writedρ =ρ(x)dx where the Radon-Nikod´ym derivativeρ(x) is called the density function. Some- times, we will write the Radon-Nikod´ym derivative in terms of the density functions ρ(x) and µ(x) instead of using the probability densitiesdρ anddµ, i.e., “dividing out” the Lebesgue measuredx, we get

f = dρ dµ=

ρ(x)

µ(x), (1.36)

where dρ = ρ(x)dx and dµ = µ(x)dx. In this sense, the Radon-Nikod´ym derivative is a ratio of density functions.

The Radon-Nikod´ym derivative has all the usual properties of a good deriva- tive: • Ifρ≪µand ν≪µ, then d(ρ+ν) dµ = dρ dµ+ dν dµ, µ−a.e. • Ifρ≪µ≪λ, then dρ dλ = dρ dµ dµ dλ, µ−a.e. • Ifρ≪µand µ≪ρ, then dρ dµ = dµ dρ −1 .

• Ifρ_≪µ, then for integrable functionsg we have

hgiρ= Z R g(x)dρ(x) = Z R g(x)dρ(x) dµ(x)dµ(x) = gdρ dµ µ .

14_{We present the theorem here for}

R, for which the theorem was first proven by Radon.

The Girsanov formula

The Radon-Nikod´ym derivative seems like a useful tool in handling probability measures, but its definition does not prescribe us a recipe at all for calculat- ing it in a given, particular situation. The Girsanov formula tells us how to do this in some specific, but very useful cases. The Girsanov formula arises naturally when looking for the solution of the Langevin dynamics (1.33). We present here a very intuitive, hand-waiving argument which shows this. For full mathematical details, we refer to reference [76].

Rewriting the Langevin dynamics (1.33), we get r

β

2 dxt−F(xt, t)dt

=dWt.

Since the right-hand side has a Gaussian distribution with mean zero and variancedt15_{, the same is of course true for the left-hand side:}

Probdxt+∇U dt=√1 2πexp −_4dtβ dxt+∇U dt 2 .

The left-hand side of this equation tells us how the Gaussian distribution is spread around dxt =F(xt, t)dt, which is precisely what would be the equa-

tion of motion for the position xt in the absence of noise. We can interpret

Probdxt−F(xt, t)dtas the transition probability to go from a positionxt

toxt+dtin a time stepdt.

For an entire trajectory ω =_{xt|t ∈ [0, τ]}, we would get the “product” of

the initial distributionρ(x) with the transition probabilities for each infinites- imal time step dt. Leaving out the normalization constants, we get for the probability density Prob[ω]:

Prob[ω] =ρ(x0) “ τ Y t=0 ” exp −_4dtβ dxt−F(xt, t)dt 2 =ρ(x0) exp −β₄ Z τ 0 dt x˙t−F(xt, t) 2 =ρ(x0) exp −β₄ Z τ 0 dtx˙2t+ β 2 Z τ 0 dtx˙tF(xt, t)− β 4 Z τ 0 dt F(xt, t)2 . This notation is of course very problematic since ˙xt, as a derivative of a Brow-

nian motion, is really not well defined. This problem can be avoided by “di- viding” by a reference probability density Prob0[ω] for which F = 0. In the

notation of the Radon-Nikod´ym derivative (1.36), we get, assuming that each

15_{This is a property of the Wiener process: the incrementsW}

t+∆t−Wt are Gaussian

distributed with mean zero and variance ∆t. In the infinitesimal case and heuristically speaking, the incrementdWt=Wt+dt−Wt thus has a variancedt.

time we start from the same initial distributionρ(x) Prob[ω] Prob0[ω] = exp _β 2 Z τ 0 dtx˙tF(xt, t)− β 4 Z τ 0 dt F(xt, t)2 ,

which is the Girsanov formula for a Langevin dynamics in the Itˆo notation. If we go to the Stratonovich interpretation, we pick up an additional term:

Prob[ω] Prob0[ω] = exp β 2 Z τ 0 dxt◦F(xt, t)−β 4 Z τ 0 dt F(xt, t)2+∂F ∂x . (1.37)

Entropy

In this chapter, we focus on the concept of entropy. It will turn out to play a central role in our study of the macroscopic world and in the emergence of irreversibility. It is by no means our goal to be very complete and very precise: we provide an introduction to the numerous discussions on entropy in the literature, and give references to more detailed works where necessary.

2.1

The trouble with entropy

Entropy is a physical concept which is used in many different contexts. The chairman of the Lockheed Martin company, while writing about the common pitfalls for business managers, describes his 27th law as follows [4]:

Software is like entropy. It is difficult to grasp, weighs noth- ing, and obeys the second law of thermodynamics; i.e. it always increases.

– Norman Augustine

The former president of Czechoslovakia and the Czech Republic illustrates the battle against the suppression of creativity and exploration under the com- munist regime as a struggle against entropy - against monotony and sameness [56]:

Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.

– V´aclav Havel

Apparently, the word “entropy” has started to live its own life, outside the realm of physics from where it first originated. Because of this wide visibility, the concept of entropy is also a vulnerable one and much of the subtlety in- volved in the physical definitions is often lost or forgotten.

Leaving all ambiguities and conceptual problems aside however, one cannot ignore the fact that the coming to terms with “entropy” has led to much scientific progress. The formulation of thermodynamics and the associated study of heat and entropy was the driving force behind the industrial revolution in the 19th century. On top of that, also a large part of the 20th century physics was initiated or guided by the concept of entropy. Albert Einstein used considerations on entropy to introduce the photon in one of his celebrated 1905 papers [40]:

... Diese Gleichung zeigt, daß die Entropie einer monochroma- tischen Strahlung von gen¨ugend kleiner Dichte nach dem gleichen Gesetze mit dem Volumen variert wie die Entropie eines idealen Gases oder die einer verd¨unnten L¨osung. [...] Hieraus schließen wir weiter: monochromatischen Strahlung von geringer Dichte [...] verh¨alt sich in w¨armetheoretischer Beziehung so, wie wenn sie aus voneinander unabh¨angigen Energiequanten [...] best¨unde.

– Albert Einstein

Einstein’s appreciation on thermodynamics, the theory that is built almost entirely on the concept of energy and of entropy, is quite clear [41]:

A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep im- pression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.

– Albert Einstein

Max Planck describes [97] his line of thoughts which lead him to the black body radiation law, starting the theory of quantum mechanics, and for which he was awarded the 1918 Nobel prize in Physics:

However, even if the radiation formula should prove itself to be absolutely accurate, it would still only have, within the signifi- cance of a happily chosen interpolation formula, a strictly lim- ited value. For this reason, I busied myself [...] with the task of elucidating a true physical character for the formula, and this problem led me automatically to a consideration of the connec- tion between entropy and probability, that is, Boltzmann’s trend of ideas; until after some weeks of the most strenuous work of my life, light came into the darkness, and a new undreamed-of perspective opened up before me. [...] On the basis of a considera- tion of this kind a specific, relatively simple combinatorial method was obtained for the calculation of the [...] entropy expression determined by the radiation law, and it brought me much-valued

satisfaction for the many disappointments when Ludwig Boltz- mann, in the letter returning my essay, expressed his interest and basic agreement with the train of thoughts expounded in it.

– Max Planck

It thus appears that entropy has played a crucial role in the development of modern physics such as quantum mechanics, even though it is not always visibly present within modern theories.