In Nature and Nature’s God I argue that the second law of thermodynamics verifies Aquinas’ claim (SCG I.13) that a sempiternal universe would require God’s sustaining causality. The second law requires that in any isolated system the entropy must increase monotonically over time until it reaches a maximum. If there is no God the universe is an isolated system in the thermodynamic sense. Therefore its entropy must increase monotonically over time. If the universe had no beginning infinite time would have already elapsed, and the entropy density of the universe would have reached a maximum value already. The universe would thus be in a state of heat death with no free energy left, contrary to fact. Thus the universe either came into existence a finite time ago or is not an isolated system, but open to entropy reducing action from outside. Since the universe is the totality of physical beings, there must be a transcendent cause for the universe’s condition that is not physical and thus not subject to the second law of thermodynamics. This is God.
In his comments on my book at the 2023 ACPA meeting, Robert Koons pointed out that it is possible to construct a mathematical function that increases montonically from time equals negative infinity to positive infinity, approaching 0 asymptotically in the negative time direction, and approaching some finite maximum value asymptotically in the positive time direction. For example, the function f(x) = ½ + x/(2|x| + 2). To show that the second law rules out an infinitely old, isolated universe, one must show that the entropy of the universe over time cannot be described by such a function.
This can be done. For the entropy value for the universe as a whole is dominated by gravitational entropy. This means that General Relativity is the appropriate context in which to think about the entropy of the universe. In gravitational terms, free energy means the capacity for structure formation and orbital motion. Maximal entropy means either collapse into a black hole, or dispersal in a space undergoing accelerated expansion, in which interaction between bodies becomes impossible due to the finite speed of causal interaction (the speed of light). This speed eventually becomes too slow to traverse the ever expanding distance between bodies.
Albert Einstein’s original application of General Relativity to cosmology attempted to model a steady state universe that could exist in a condition like the universe’s present one for an infinity of time past and future. But it quickly became clear that this was not possible. Such a universe is at a point of unstable equilibrium, and must either begin contracting and collapsing, or begin a runaway expansion at the slightest disturbance.
Applying this to the shape of possible entropy time functions, gravitational entropy cannot increase at an arbitrary slow rate, as an infinitely old universe subject to the second law would require. Entropy has a discrete component, and possibly a continuous one as well. The discrete component is constituted by the quantum states of a system. Entropy is Boltzmann’s constant times the logarithm of possible microstates that could instantiate a given macrostate. (S=k ln W). A minimum of entropy would be zero, representing a single microstate. (log 1=0). The smallest possible increase of entropy would be k ln 2, representing two possible microstates. Thus the entropy would have to increase at a minimal finite rate, which would require only a finite time to reach the present level of entropy.
Currently, however, the energy of a body is accounted for not only by its discrete quantum states, but by its overall translational kinetic energy due to its motion through space. Space and time are not quantized in General Relativity or classical mechanics, and there is no established quantum theory of gravity or space and time. To apply Boltzmann’s entropy formula to this aspect of physical systems, one establishes a measure for the volume of phase space that a system in a given macrostate can occupy. Then the W in S=k ln W is not an integer value of microstates, but a volume in phase space. In principle the entropy value becomes infinitely divisible. (If time, space, and gravity turn out to be quantized and discrete, as many people think, the case for a minimal rate of entropy increase is strengthened. But physicists have been looking for a theory of quantum gravity for a hundred years and have remained unsuccessful. I believe time and space just are fundamentally continuous.)
However, the Heisenberg uncertainty principle should still apply, according to which there is some finite, minimum level of uncertainty in the position and momentum of a physical body or system. This indeterminacy will set a lower limit on how finely-tuned the gravitational system of the universe can be, and thus a lower limit on the rate of collapse/dispersal, and thus a lower limit on the rate of entropy increase for the universe overall.
Thus it is not possible for the entropy of the universe to increase monotonically for an infinity of time past and future. And thus the universe must have been started by God a finite time ago, or be sustained by God in a low entropy condition. Either way, naturalism is false.