The relentless march of randomness

The Second Law of thermodynamics – what law is this? What is its central role in physical behaviour? And in what way does it present us with a genuinely deep mystery? In the later sections of this book, we shall try to understand the puzzling nature of this mystery and why we may be driven to extraordinary lengths in order to resolve it. This will lead us into unexplored areas of cosmology, and to issues which I believe may be resolved only by a very radical new perspective on the history of our universe. But these are matters that will be our concern later. For the moment let us restrict our attention to the task of coming to terms with what is involved in this ubiquitous law.

Usually when we think of a ‘law of physics’ we think of some assertion of equality between two different things. Newton’s second law of motion, for example, equates the rate of change of momentum of a particle (momentum being mass times velocity) with the total force acting upon it. As another example, the law of conservation of energy asserts that the total energy of an isolated system at one time is equal to its total energy at any other time. Likewise, the law of conservation of electric charge, of momentum, and of angular momentum, each asserts a corresponding equality for the total electric charge, for the total momentum, and for total angular momentum. Einstein’s famous law E= mc² asserts that the energy of a system is always equal to its mass multiplied by the square of the speed of light. As yet another example, Newton’s third law asserts that the force exerted by a body A on a body B, at any one time, is always equal and opposite to the force acting on A due to B. And so it is for many of the other laws of physics.

These are all equalities – and this applies also to what is called the First Law of thermodynamics, which is really just the law of conservation of energy again, but now in a thermodynamic context. We say ‘thermodynamic’ because the energy of the thermal motions is now being taken into account, i.e. of the random motions of individual constituent particles. This energy is the heat energy of a system, and we define the system’s temperature to be this energy per degree of freedom (as we shall be considering again later). For example, when the friction of air resistance slows down a projectile, this does not violate the full conservation law of energy (i.e. the First Law of thermodynamics) – despite the loss of kinetic energy, due to the projectile’s slowing – because the air molecules, and those in the projectile, become slightly more energetic in their random motions, from heating due to the friction.

However, the Second Law of thermodynamics is not an equality, but an inequality, asserting merely that a certain quantity referred to as the entropy of an isolated system – which is a measure of the system’s disorder, or ‘randomness’ – is greater (or at least not smaller) at later times than it was at earlier times. Going along with this apparent weakness of statement, we shall find that there is also certain vagueness or subjectivity about the very definition of the entropy of a general system. Moreover, in most formulations, we are led to conclude that there are occasional or exceptional moments at which the entropy must be regarded as actually (though temporarily) reducing with time (in a fluctuation) despite the general trend being that the entropy increases.