Fashion, Faith, and Fantasy in the New Physics of the Universe by Roger Penrose

Author:Roger Penrose
Language: eng
Format: epub, pdf
Publisher: Princeton University Press
Published: 2016-10-14T16:00:00+00:00

However, the number of submicroscopic states N12 that go to make up Σ1 and Σ2 together would be the product N1N2 of the number N1 that go to make up Σ1 and the number N2 that go to make up Σ2 (since each of the N1 ways of making up Σ1 may be accompanied by any one of the N2 ways of making up Σ2). To convert the product N1N2 to the sum S1 + S2, all we need to do is to use a logarithm in the definition of the entropy (§A.1):

where we choose some convenient constant k.

This is, indeed, essentially the famous definition of entropy given by the great Austrian physicist Ludwig Boltzmann in 1872, but there is one further point that needs to be made clear in this definition. In classical physics, the number N is normally going to be infinite! In accordance with this issue, we really have to think of this “counting” in a rather different (and more continuous) way. In order to express this procedure succinctly, it is best that we return to the notion of phase space, which was introduced in its essentials in §2.11 (and is explained more fully in §A.6). We recall that the phase space P, of some physical system, is a conceptual space, normally of a very large number of dimensions, each of whose points represents a complete description of the submicroscopic state of the (say, classical) physical system being considered, this state encompassing all the motions (as given by their momenta) as well as all the positions of all particles that constitute the system. As time evolves, the point P, in P, which represents the submicroscopic state of the system, will describe a curve C in P, whose location within P will be fixed by the dynamical equations, once the location within P of any particular (initial) point P0, on C, has been chosen. Any such point P0 will fix which actual curve C gives us the time evolution of our particular system (see figure A-22 of §A.7) described by P (where P0 describes the initial submicroscopic state of the system). This is the nature of the determinism that is central to classical physics.

Now, in order to define the entropy, we need to collect together – into a single region called a coarse-graining region – all those points in P which are considered to have the same values for their macroscopic parameters. In this way, the whole of P will be divided into such coarse-graining regions; see figure 3-12. (We may have to think of these regions as having fairly “fuzzy” boundaries, as there will always be slightly problematic issues of defining precisely where the boundaries of these coarse-graining regions actually are.) The points of P that lie in the neighbourhoods of such boundaries are normally considered to be an insignificant fraction of the total, and so can be ignored. (See §1.4 of Penrose [2010], especially figure 1.12.) Thus, the phase space P will be divided up into

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