## 3psilon 3 years ago Why does it take infinite time to achieve final equilibrium ?

• This Question is Open
1. Carl_Pham

Who says it does?

2. 3psilon
3. 3psilon

Page 34 5th paragraph

4. Carl_Pham

I see. This is sort of a silly statement. It probably makes the assumption that, because the driving force for any return to equilibrium is, roughly speaking, equal to the "distance" of the system from equilibrium, the driving force will steadily decrease to zero as the system approaches equilibrium, so that the approach of the system to equilibrium is asymptotic. For example, the rate of heat flow out of an object heated above ambient is proportional to the difference in temperature between the object and the surroundings. As heat leaves the object, the "driving force" for heat flow (the temperature difference) declines, so the rate of heat flow falls, and the rate of cooling declines. You might imagine, therefore, that the time ti takes the object to reach EXACTLY ambient temperature is infinite, although it will get as close as you like in a finite time you can calculate, e.g. it wil be within 1 degree in 10 minutes, say, and therefore within 0.1 degree in 100 minutes, 0.01 degrees in 1,000 minutes, and so forth. The reason this is silly is that it forgets about fluctuations, which are an important phenomenon even within classical thermodynamics, as here. Fluctuations means a system is always wandering a bit from the values of the thermodynamic variables that define "the" equlibrium state. For example, an object in thermal equilibrium with its environment will, nevertheless, always experience tiny fluctuations in temperature above and below ambient, as tiny bits of excess heat energy flow into and out of the object. These fluctuations necessarily put a lower bound on how close a system can be to equilibrium and still be described as not in equilibrium. That is, if you are considering a system cooling to thermal equilibrium, you can no longer logically say it is not in equilibrium when the average tempature difference between the object and the environment is less than or equal to the average fluctuation in temperature. That will always occur in a finite amount of time -- so equilibrium does not, in fact, take an infinite amount of time. These fluctuations always exist, and in fact there is a deep connection between the relaxation of the system to equlibrium and the size of the fluctuations at equilibium, called the Fluctuation-Dissipation Theorem.

5. Carl_Pham

I will say, however, that it is important to understand that approaches to equilibrium generally do have an asymptotic character. The reason is as I said above, to first order, the size of the driving force for change is proportional to the difference between the value of some order parameter in the system now and when it is in equilibrium. As the system approaches equlibrium, this driving force necessarily vanishes. A very general result is that you can expect an exponential decay of the order parameter to its equilibrium value. So, for example, you should expect the temperature of a hot object cooling, the pressure of a system leaking, et cetera, to fall like exp(-t/T) where t is time and T is some characteristic time that will depend on the details of the process.

6. 3psilon

Wow thank you for such s detailed explanation ! I just started thermo and you made it pretty clear ! Thank you so much @Carl_Pham

7. Carl_Pham

Ah, the queen of empirical science. See if you can keep a long perspective on thermo. It's easy to get caught up in its thickets of mindless equations about ideal gases and isothermal this and that, and think it's just a dull piece of rigamarole. But in fact thermodynamics has an astounding simplicity, generality, and elegance which will richly reward you if you can get past all that. I wish I could think of a good Stephen Hawking like overview that would convey this impression, but thermodynamics is so far out of favor in modern instruction that's tough. H. B. Callen's classic graduate textbook ("Thermodynamics") as some, but it's very dry. You can probably get a taste by googling terms like "entropy of the universe," "arrow of time," "Maxwell's Demon," "Gibbs Paradox," and "Poincarre recurrence."

8. 3psilon

9. UnkleRhaukus

equilibrium is not equilibrium

10. JeanetteBaker

this is probably a more theoretical question