Entropy and the Second Law of Thermodynamics are thought to be constructs that only apply on a macro level and for system involving a number of particles on the order of millions and greater. It is argued here that these concepts apply on the nuclear and subnuclear levels.

The Dimensions of Entropy

The change in entropy ΔS for a system at temperature T which experiences a transfer of Q units of heat energy of Q is

ΔS = Q/T

The temperature is the average energy per particle in the system. Let E be the total heat energy of a system of N particles. Then

T = E/N

Thus

ΔS = (Q/E)N

Since Q and E are in units of energy, entropy has no physical units; it is proportional to the number of particles in the system.

This is in contrast to the result from Statistical Mechanics that entropy is given by:

S = k_Blog(Ω)

where k_B is Boltzmann's constant and Ω is the number of states of the system.

The Second Law of Thermodynamics

The Second Law of Thermodynamics is that the entropy of a closed system does not decrease. For particle and nuclear interactions this means that the number of particles coming out of an interaction is greater than or equal to the number going into that interaction. The count of particles has to include photons as well as material particles.

For the Second Law it does not matter whether entropy is equal to the number of particles or the logarithm of the number of particles or any other monotonically increasing function of the number of particles. In any such case entropy does not decrease if the number of particles does not decrease.

Illustrations of the Second
Law in Particle Interactions

Consider the case of the decay of a neutron

n → p + γ

where γ stands for a gamma ray, p is for a proton and n for a neutron. Thus one particle is transformed into two particles.

The fission of uranium 235 also beautifully illustrates the case.

U₂₃₅ + n → Ba₁₄₄ + Kr₈₉ + 3n + γ

If the nuclei of U₂₃₅, Ba₁₄₄ and Kr₈₉ count as only one particle each then this case involves two particles being transformed into six particles, counting the photon. However if a nuclide is composed of substructures such as alpha particles, nucleon pairs and singleton nucleons the count is more complicated. The barium and krypton nuclei probably preserve the substructures of the uranium nucleus but there is an increase of two neutrons and one photon in the reaction.

Not all isotopes of uranium fission. For example,

U₂₃₈ + n → Np₂₃₉ + e⁻ + ν_e
and
Np₂₃₉ → Pu₂₃₉ + e⁻ + ν_e

where ν_e stands for an electron anti-neutrino.

In the first reaction, the creation of neptunium 239, there is an increase in the number of particles of one and likewise in the decay of the neptunium into plutonium, for an overall increase of two particles.

Of course it is easy to display numerous instances of the number of particles increasing as a result of nuclear fission. The real test is of the annihilation of particles.

Consider the mutual annihilation of an electron and a positron.

e₋ + e₊ → 2γ

Two gamma rays are emitted to preserve the momenta of the electron and positron. Here two particles are replaced by two particles.

If an electron and positron meet in the vicinity of a nucleus the momentum is conserved partially by the adjacent nucleus. In this case only one gamma ray is emitted. Initially there are three particles but after the annihilation there are only two. However the positron is the anti-particle of the electron. This case suggests that anti-particles should count as negative particles. Thus the initial count of the electron-positron-nucleus system is 1=(1−1+1) and after the annihilation it is 2=(1+1). In the case of the two gamma ray annihilation the net count goes from zero to two.

The formation of a deuteron, a neutron-proton pair, is an interesting case.

p + n → d + γ

In this case the count is two before and after the interaction.

Thus the Second Law of Thermodynamics appears to hold also on a nuclear and subatomic particle level. This is an overlooked instance of the Second Law.

There is one apparent contradiction of this proposed "law" but the incidence of this contradiction is so small that it is almost a confirmation. When a positron and electron annihilate normally there are two photons produced traveling in opposite directions. This allows the conservation of momentum and energy. When a positron encounters an electron in the K shell of a heavy nucleus, say caesium, then only one photon is created and the recoil of the nucleus preserves momentum and energy. This type of annihilation occurs only about 0.2 of 1 percent of the time as documented in the article "Single-Quantum Annihilation of Positrons" in Physical Review (Dec. 1961) (pages 1851-1861) by Lester Sodickson et al.

And this phenomenon may not involve a violation of the Second Law. Some amount of the energy from the masses of the electron and positron is transformed into kinetic energy for the associated nucleus involved in the single-quantum annihilation. In effect this is heat energy, say ΔQ. The increase in entropy for the associated nucleus is then ΔQ/T, where T is the temperature of the associated nucleus. But temperature is in the nature of kinetic energy per particle and can be so expressed using Boltzmann's constant k_B. Therefore the increase in entropy for the associated nucleus is ΔQ/(k_BT) and this is in units of particles. So the one particle lost in the single-quantum annihilation can be more than made up for in the increase in entropy for the associated nucleus for such nuclei at low temperatures..