Question

Your task related to this topic is to work out what the analogous“linear transport relation”should be for describing chemistry in the Onsager picture. You need to identify the “force,”the “flux,”and the “transport coefficient.”No need to worry about how chemistry would interact with other forces and/or fluxes, just an expression describing a chemical reaction itself as a “dissipative channel”that would represent a diagonal term in the matrix of Onsager transport relations (the off-diagonal terms would describecoupling between different forces and fluxeswhile the diagonal terms are all the traditional equations that came before Onsager). Correct answers will refer to single chemical reactions, bonus points if you can write the expression fora chemical reaction network with a generic number of reactions taking place inside it. There is one point that I need to notebefore you get started with this problem, however. The problem is almost as simple as just looking at the three equations relating flux and force, figuring out which state variables would be used for chemistry (hope that’s obvious but I gave the answer in the notes above if not) and then deciding whether you want to use a conductance or resistance coefficient and just “writing it down.”There is one complication you will need to puzzle out, however, which is probably also the reason that there isn’t a law of this sort already defined for chemical reactions. I’ll give you a hint: it has to do with themeaning of the statement chemistry teachers say“don’t confuse thermodynamics with kinetics,”which I told you really meant “don’t confuse a barrier height with a well depth,”but we could also say “don’t confuse equilibrium thermodynamics with kinetics.”OK, that’s the only complication that makes the answer to the question atall different than the other three equations and once you get it then you should be able to just write downthe answer.

Answer 1

B

OLTZMANN transport equation has played an important role in basic and applied sciences.

It is a nonlinear integro-differential equation for the phase space density of the molecules of

a dilute gas. It remains today, an important theoretical technique for investigating non-equilibrium

systems. It was derived by Ludwig Eduard Boltzmann (1844 - 1906) in his further studies on thermal equilibrium between gas molecules [1], published in the year 1872. Boltzmann did this work

solely for purpose of addressing the conflict between time-reversal-invariant Newtonian mechanics

and time-arrowed thermodynamics. Linear version of this equation [2] provides an exact description of neutron transport in nuclear reactor core and shields. Linear transport equation constitutes

the backbone of nuclear industry. It is indeed appropriate that the Indian Society for Radiation

Physics (ISRP) has chosen Boltzmann transport equation as focal theme for the sixteenth National

Symposium on Radiation Physics (NSRP-16), in Meenakshi College for Women, Chennai during

January 18 - 21, 2006. The year 2006 marks the hundredth anniversary of Boltzmann’s death.

There are going to be several talks [3] in this symposium, covering various aspects of linear

transport equation. However, in this opening talk, I shall deal with nonlinear transport equation.

I shall tell you of Boltzmann’s life-long struggle for comprehending the mysterious emergence

of time asymmetric behaviour of a macroscopic object from the time symmetric behaviour of its

microscopic constituents. In the synthesis of a macro from its micro, why and when does time

reversal invariance break down? This is a question that haunted the scientists then, haunts us now

and most assuredly shall haunt us in the future, near and far.

The Second law is about macroscopic phenomena being invariably time asymmetric; it is about macroscopic behaviour being almost always irreversible 1

. Physicists think the Second law can

not be derived from Newton’s equations of motion. According to them, the Second law must be a

consequence of our inability to keep track of a large number, typically of the order of 1023 or more,

of molecules. In other words, the origin of the Second law is statistical. It is one thing if statistics

is used merely as a convenient descriptor of a macroscopic phenomenon. It is quite another thing

if we want to attribute an element of truth to such a description. Is it conceivable that nature is

deterministic at micro level and stochastic at macro level? Can (microscopic) determinism give

rise to (macroscopic) unpredictability? Boltzmann thought so.

Boltzmann believed that the Second law is of dynamical origin. He proved it through his transport equation and H-theorem. At least he thought he did. Several of his fellow men thought

otherwise. It is this fascinating story of the Second law that I am going to narrate to you in this talk.

I am going to tell you of the insights that Boltzmann provided through his early work on transport

equation and his later work that laid the foundation for Statistical Mechanics - a subject that aims to

derive the macroscopic properties of matter from the properties of its microscopic constituents and

their interactions. I am also going to tell you of nonlinear dynamics and chaos, subjects that have

completely changed our views about determinism, dynamics and predictability. Now we know that

determinism does not necessarily imply predictability. There are a large number of systems that

exhibit chaotic behavior. Chaos and hence unpredictability is a characteristic of dynamics. Thus,

Boltzmann’s hunch was, in essence, right. It was just that he was ahead of his time.

Boltzmann staunchly defended the atomistic view. He trusted atoms [4]. He was of the opinion

that atomistic view helps at least comprehend thermal behaviour of dilute fluids. But the most

1Deterioration, dissipation, decay and death characterize macroscopic objects and macroscopic phenomena. A

piece of iron rusts; the reverse happens never. A tomato rots, inevitably, invariably and irreversibly. An omelet is easily

made from an egg; never an egg from an omelet.

The physicists are puzzled at the Second law. How does it arise ? An atom - the constituent of a macroscopic object,

obeys Newton’s laws. Newtonian dynamics is time reversal invariant. You can not tell the past from the future; there is

the determinism - the present holding both, the entire past and the entire future. The atoms, individually obey the time

reversal invariant Newtonian dynamics; however their collective behaviour breaks the time symmetry.

The philosophers are aghast at the implications of the Second law. Does it hold good for the Creator ? They are

upset at the Second law since it spoils the optimism and determinism implicit in for example in the verse below from

Bhagavat Gita, an ancient text from the Hindu Philosophy:

Whatever happened, it happened

for good.

Whatever is happening, is

happening for good.

Whatever that will happen, it will

be for good.

Omar Khayyam surrenders to the irreversibility of life when he writes,

The Moving Finger writes; and, having writ,

Moves on: nor all your Piety nor Wit

Shall lure it back to cancel half a Line,

Nor all your Tears wash out a Word of it.

Bernard Shaw, frustrated with the Second law, exclaims youth is wasted on the young. Mark Twain hopes fondly for

Second law violation when he wonders life would be infinitely happier if only we could be born at eighty and graduallyinfluential and vociferous of the German-speaking physics community - the so-called energeticists,

led by Ernst Mach (1838 - 1916) and Wilhelm Ostwald (1853 - 1932) did not approve of this.

For them, energy was the only fundamental physical entity. They dismissed with contempt any

attempt to describe energy or transformation of energy in more fundamental atomistic terms or

mechanical pictures. This lack of recognition from the members of his own community allegedly

led Boltzmann to commit suicide 2

. Ironically, Boltzmann died at the dawn of the victory of the

atomistic view. For, in the year 1905, Albert Einstein (1879 - 1955) established unambiguously the

reality of atoms and molecules in his work [5] on Brownian motion.

2 On the nature of things

It all started with our efforts to understand the nature of matter, in general and of heat, in particular.

Ancient man must have definitely speculated on the possibility of tiny, invisible and indivisible

particles assembling in very large numbers into a visible continuum of solids and liquids and an

invisible continuum of air that surround us. The Greeks had a name for the tiny particle: atom - the

uncuttable. According to Leucippus (440 B.C.) and his student Democritus (370 B.C.) atom moves

in void, unceasingly and changing course upon collision with another atom. Titus Lucretius Carus

(99 B.C. - 55 B.C.) mused on the nature of things 3

. According to him all the phenomena we see

around are caused by invisible atoms moving hither and thither 4

. There was no role for God in his

scheme of things. Atomism of the very early times was inherently and fiercely atheistic. Perhaps

this explains why it lost favour and languished into oblivion for several centuries.

3 Revival of Atomistic view

The revival came with the arrival of Galileo Galilei (1564-1642) who wrote in the year 1638, of the

air surrounding the earth and of its ability to stand thirty four feet of water in a vertical tube closed

at the top with the open bottom end immersed in a vessel of water. He also knew of air expanding upon heating and invented a water-thermo-graph (thermometer). A few years later, his student

2Boltzmann enjoyed the respect of all his colleagues. Rejection of his ideas by the energeticists does not seem to be

the only reason or even one of the reasons that drove him to his tragic end. Men like myths. Men like heroes. Scientists

are no exception. Scientists need heroes - tragic or otherwise. Boltzmann is one such.

3Lucretius wrote a six books long poem called De Rerum Natura (On the Nature of Things) on atomism.