# Is there any relation between the entropy of a system and the kinetic energy of molecules in the system?

The kinetic energy of the molecules in the system increases when we provide heat to the system,thereby increasing its temperature under certain conditions.

Similarly providing heat to a liquid increases the kinetic energy of liquid molecules and hence it's entropy to an extent.

But at the Big Bang the temperature is infinitely very much,so will be the kinetic energies of the molecules at that time,but at the Big Bang the universe is said to be in a state of high order.Is there any relation between kinetic energy and entropy?

Entropy can be understood as the degree to which a system's microstate (the details of exactly what all its component parts are doing) is not fixed by the constraints imposed by the system's surroundings. In the case of a gas (which is a good way to think of the early universe, ignoring the fact that it is really a plasma) the microstate is fixed by specifying the combination of position and momentum for each particle. We can then measure the entropy by the area of the region of a momentum/position plot that is filled by the possible states of the motion of all the particles. The following diagram shows four example cases. In each plot, the ellipse is intended to show, approximately, the range of position and momentum values of the gas particles.

I have labelled each example with a comment on temperature $$T$$ and density $$\rho$$. A low spread of momentum ($$p$$) values indicates a low temperature. A low spread of position ($$x$$) values indicates a high density. The area of the ellipse indicates the entropy. The four cases are self-explanatory (I hope).

It is true that, other things being equal, a high temperature will lead to a high entropy, because the range of $$p$$ values increases. However, it does not always happen because a high density brings the range of $$x$$ values down, and this lowers the entropy again.

The early universe is an example of case D.

The subsequent evolution of the universe is largely a movement from case D to case C. This is called an adiabatic expansion, in which the entropy does not change, even though the temperature falls.

To summarise, there is no general relation which says that hot things should have either high or low entropy, if they can also be dense. High density tends to bring the entropy down.

Finally, a comment on the early universe. There is no need to bring in the word 'infinity'. All we know is that the very early universe was very hot and very dense. At sufficiently early times it was in a parameter regime where all our knowledge of physics runs out, but this does not mean we know it was infinite in any respect (whether in terms of kinetic energy or density or volume or other such parameters).

• Very good and interesting answer. Thanks! – Cham Aug 4 '19 at 14:40

In an isolated system only the potential energy and the entropy energy matters. Initially the system can be cold so be at 0K. However, the potential energy of the system is not zero and it will start to relax down to the potential energy landscape performing work, lowering potential energy and increasing the entropy of the system (increasing kinetic energy or temperature). So one can ask, if we have only one particle lets say located on a slope of a parabolic valley. What will be the entropy and actually how one can define it? If we consider the particle then entropy is minimal of the initial state where particle was at the potential energy maximum and was not moving. Particle occupied, in this state, only one spot. However, when it moves in the valley with any kinetic energy, even behaving as a lossless harmonic oscillator (converting potential energy into kinetic an vise versa), the entropy of this state of the particle will be higher as in space coordinates the particle is not anymore localized but rather occupies some place in average. So the potential energy in average has been lowered as well as entropy increased, so leading to a more favorable state. So the pressure, the particle exerts in average on walls of the valley, one can define as an analogy of temperature which obviously depends on the shape of the valley and the initial potential energy of the particle.