(1) *For an isolated system, the direction of spontaneous change is from an arrangement of lesser probability to an arrangement of greater probability.*

(2) *For an isolated system, the direction of spontaneous change is from order to disorder.*

Like the conservation laws, the second law of thermodynamics applies only to a system free of external influences. For a system that is not isolated, there is no principle restricting its direction of spontaneous change.

Another statement of the second law of thermodynamics makes use of the concept of *entropy*. Entropy measures the extent of disorder in a system, or, equivalently, the probability of the arrangement of the parts of a system. For an arrangement of greater probability, which means greater disorder, the entropy is higher. An arrangement of less probability (greater order) has less entropy. Here is a statement of the second law in terms of entropy:

(3) *The entropy of an isolated system increases or remains the same.*

Specifically, entropy, for which the usual symbol is *S,* is defined as Boltzmann’s constant multiplied by the natural logarithm of the probability of any particular state of the system:

*S *= *k *log *P.*

The appearance of Boltzmann’s constant *k *as a constant of proportionality is a convenience in the mathematical theory of thermodynamics, but is, from a fundamental point of view, entirely arbitrary. The important aspect of the definition is the proportionality of the entropy to the logarithm of the probability *P*. Note that since the logarithm of a number increases when the number increases, greater probability means greater entropy, as stated in the preceding paragraph.

Since the velocities as well as the positions of individual molecules are generally unknown, velocity, like position, is subject to considerations of probability. This kind of probability, like the probability of position, follows the rule of spontaneous change from lower to higher probability. It should not be surprising to learn that for a collection of identical molecules the most probable arrangement is one with equal average speeds^{1} (and randomly oriented velocities). This means that available energy tends to distribute itself uniformly over a set of identical molecules, just as available space tends to be occupied uniformly by the same molecules. In fact, the equipartition theorem and the zeroth law of thermodynamics can both be regarded as *consequences *of the second law of thermodynamics. Energy divides itself equally among the available degrees of freedom, and temperatures tend toward equality, because the resulting homogenized state of the molecules is the state of maximum disorder and maximum probability. The concentration of all of the energy in a system on a few molecules is a highly ordered and improbable situation analogous to the concentration of all of the molecules in a small portion of the available space.

The normal course of heat flow can also be understood in terms of the second law. Heat flow from a hotter to a cooler body is a process of energy transfer tending to equalize temperature, which increases entropy (because it increases disorder). The most probable distribution of the energy in a system is its equal division among degrees of freedom (equipartition), just as the most probable spatial arrangement of molecules is equal likelihood of being anywhere.

Heat flow is so central to most applications of thermodynamics that the second law is sometimes stated in this restricted form:

(4) *Heat never flows spontaneously from a cooler to a hotter body.*

Notice that this is a statement about macroscopic behavior, whereas the more general and fundamental statements of the second law, which make use of the ideas of probability and order and disorder, refer to the submicroscopic structure of matter.

Historically, the first version of the second law, advanced by Sadi Carnot in 1824, came before the submicroscopic basis of heat and temperature was established, in fact before the first law of thermodynamics (energy conservation) was formulated. Despite a wrong view of heat and an incomplete view of energy, Carnot was able to advance the important principle that no heat engine (such as a steam engine) can operate with perfect efficiency. In modern terminology, Carnot’s version of the second law is this:

(5) *In a closed system, heat flow out of one part of the system cannot be transformed wholly into mechanical energy (work), but must be accompanied by heat flow into a cooler part of the system. *

In brief, heat cannot be transformed completely to work.

Of these five versions of the second law, the first three, expressed in terms of probability, of order-disorder, and of entropy, are the most fundamental. Worth noting in several of the formulations is the recurring emphasis on the negative. Entropy does *not *decrease. Heat does *not *flow spontaneously from a cooler to a hotter region. Heat can *not *be wholly transformed to work. A sixth version is also expressed in the negative:

(6) *Perpetual-motion machines cannot be constructed.*

This statement may sound more like a staff memorandum in the Patent Office than a fundamental law of nature. It may be both. In any event, it is certainly the latter, for from it can be derived the spontaneous increase of probability, of disorder, or of entropy. It is specialized only in that it assumes some friction, however small, to be present to provide some energy dissipation. If we overlook the nearly frictionless motion of the planets in the solar system and the frictionless motion of single molecules in a gas, everything in between is encompassed.

1^{} Actually, the root-mean-square average.