# If entropy decreases for cold systems, isn't the heat death of the universe a state of low entropy? [duplicate]

Entropy is a consequence of heat. The heat death of the universe results in an approach to absolute zero temperature. Does this mean the end of the universe is low entropy?

If entropy decreases for cold systems,

There is some misunderstanding in this statement, using the verb "decreases". from the three laws of thermodynamics entropy either remains constant on increases.

If one compares a cold system, to the same system at higher temperatures, the entropy is higher in the hotter system , but in order to get a hot system to a cold state a larger combined system is needed in order to obey the thermodynamic law 2:

Second law of thermodynamics: In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. Equivalently, perpetual motion machines of the second kind (machines that spontaneously convert thermal energy into mechanical work) are impossible.

Entropy as a variable is connected to temperature through differential equations, not linear ones, so it does not necessarity mean "low temperature"= " low entropy" as you assume

It can be demonstrated that all definitions of entropy are equivalent mathematically.

isn't the heat death of the universe a state of low entropy?

The useful definition of entropy for cosmological purposes , comes from statistical mechanics

Specifically, entropy is a logarithmic measure of the number of states with significant probability of being occupied

Entropy is a consequence of heat. The heat death of the universe results in an approach to absolute zero temperature. Does this mean the end of the universe is low entropy?

Consequence and linear dependence are two different concepts. Heat generates microstates and thus increases entropy, but it is not the only variable that can increase entropy.

So it is possible to have a system of particles in a very large volume with very small kinetic energy ( which is directly connected to temperature) to a huge number of states occupied by these particles, so that the entropy is large but the temperature close to zero.

• How many states are there at the heat death of the universe? – Hierarchist Jun 3 '19 at 22:53
• @Hierarchist a very large number, all of the quantum mechanical permutations in addition to all the space permutations of the particles involved. – anna v Jun 4 '19 at 2:56

First of all, I agree with @anna v answer. I just want to expand (no pun intended) on the last paragraph.

Entropy is related to the dispersion of energy. In that sense, it is related to the universe gradually moving to its most probable state on order that all the energy is equally dispersed throughout the universe. At the present time energy is concentrated in particular areas (you might some of them "hot spots") but as the theory goes it will eventually spread out all over the universe (whatever all over means) because that is the most probable state.

Now as the temperature of the universe goes down there is less energy available to do work. A the present time we take advantage of temperature differences and the fact that heat flows from high to low temperature by taking some of that heat from the high temperature body and doing work in a cycle, while necessarily rejecting some of it to the lower body due to the second law. The energy rejected to the low temperature body is no longer available to do work in the current environment. We would need to take our heat engine and operate it between the lower temperature body and yet another even lower temperature body. Each time we do this we have less of the original heat energy available for work since our engine can never be 100% efficient. What's more we need to continue seeking lower temperature bodies to reject heat to.

Eventually as we approach absolute zero we will no longer be able to find a lower temperature body. In other words, the temperatures will be equal and we will no longer be able to do work.

Hope this plus anna v's answer helps.

• So at the heat death of the universe, there will be no other possible state to transition to, thus there is only one state of the heat death? – Hierarchist Jun 3 '19 at 22:54
• @Hierarchist If the universe is in thermal equilibrium there are no differences in temperature between regions. Heat is energy transfer due solely to temperature difference. Without temperature difference there can be no energy transfer in the form of heat and therefore no heat flow to drive heat engines to do work. – Bob D Jun 4 '19 at 0:16
• So it's in a single state. – Hierarchist Jun 4 '19 at 0:35
• @Hierarchist OK, but I think I would call it the single most probable state. – Bob D Jun 4 '19 at 0:40
• But anything else would be not the heat death, and there are infinitely more not-heat deaths than there are the one heat death, right? – Hierarchist Jun 4 '19 at 1:49