Re: Is time required for a body to attain perfect equilibrium finite or infinite?

Dear Vineet,

Thank you for your interesting question concerning the time-dependence 
associated with achieving thermal equilibrium.  We have to consider what 
equilibrium means and how that definition changes for the system at hand.

Most thermodynamic textbooks begin with discussions of the ideal gas law 
and of single component systems in "closed systems".  In general, these 
are adiabatic systems useful for introducing the basic laws of 
thermodynamics in their simplest forms.  This sort of system is by 
definition at equilibrium, with a defined T, V, N, and corresponding 
defined thermodynamic parameters.  Of course, ideal systems are not real 
systems as they generally ignore the second law of thermodynamics (dS > 
0).  The key concept for us is that Temperature is a bulk property of any 
system that reflects its internal energy; fundamentally your question is 
about a system coming to an energetic equilibrium once two objects are 
connected.

In your example, we are concerned with measuring the temperature of 
different positions on a rod that is in contact with a heat source on one 
end, and the change of temperature as a function of time and distance from 
the source.  You have correctly surmised that the heat flow and thus 
equilibrium has directionality.  You have also correctly surmised that the 
time required to reach equilibrium is significant and system dependent.

Let's start by putting this scenario into an "ideal system"; both the body 
and rod are made of infinitely conductive material and are in a vacuum.  
When connected, the body transfers a finite quantity of heat to the rod, 
and both achieve an equilibrium temperature that is constant and uniform.  
If the mass of the body is effectively infinitely large compared to the 
rod, the temperature of the body will be virtually unchanged, and the rod 
will adjust temperature to that of the body.  If we use real materials 
(that have realistic conductivity), then equilibrium will be achieved in 
the same way but will not be instantaneous.  The conductance of the 
material will determine the rate of heat transfer, as well as the 
length/diameter/shape of the rod, the mass of both objects, etc.

If we are still in a vacuum, then at some finite time the system will come 
to a thermal equilibrium with all points of the rod at the same 
temperature as the body.  Your intuition suggests an asymptotic approach 
to equilibrium that has in infinite endpoint in time, but at some finite 
point the uncertainty in thermal equilibrium will be equivalent to the 
statistical uncertainty in the thermodynamic definition of internal energy 
and temperature...that's what's really meant by reaching equilibrium.

If we consider body (at T = 100C) and room temperature rod (T ~ 22C) and 
remove them from a vacuum and place them in a open lab environment, then 
there is another heat sink, that of the room.  Now the system will come to 
an equilibrium position in some finite amount of time; however, the rod 
will have different temperatures at different positions along its length.  
Qualitatively, the rate of heating from the body will come to an 
equilibrium with the rate of heat loss to atmospheric gasses, giving rise 
to a small heat gradient at equilibrium across the length of the rod.

Whether presented in their basic forms as in a Physical Chemistry textbook 
or in their statistical thermodynamic expression presented in advanced 
graduate-level textbooks, thermodynamic functions turn out to be 
independent of the "ensemble" used in the calculation (in other words, 
whether talking about pressure-volume work or thermal work, the value of 
internal energy has the same thermodynamic meaning).  The same is not true 
for fluctuations around the equilibrium position, nor is it true of the 
time it takes to reach equilibrium.  For each environment (and therefore 
for each ensemble considered for a particular environment), the variables 
themselves that fluctuate are system dependent.  

Taking an example from "An Introduction to Statistical Thermodynamics" by 
Terrel Hill, we can describe the energy of our ideal closed system 
immersed in a large heat bath (N, V, T are defined and constant).  Noting 
that N and V are constant, variations in system energy, E, must be 
associated with heat exchange between system and bath (defined in 
statistical thermodynamics as a "canonical ensemble").  The probability 
distribution for different energies is gaussian in shape about the mean 
value ; the dispersion or standard deviation about the mean is defined 
as:

std. dev. = (|E - |)^0.5

Hill goes on to show that the right half of the equation may be 
differentiated with respect to the microscopic energy levels at constant 
(N,V) and as a function of variable kT, which ultimately gives the 
following definition of fluctuations in E around :

E^2 - ^2 = (std. dev.)^2 = (kT^2)Cv

At any temperature, the system AT EQUILIBRIUM experiences thermal 
fluctuations with a Gaussian-distributed fluctuation around the mean, 
determined by the boltzman constant and the heat capacity at constant 
volume.  Your real-world example may be similarly described in a 
quantitative sense, taking into account the physical description of the 
rod and body (shape, density, conductance, etc.) along with the rate of 
heat transfer to air, the positional dependence, etc.  However, the system 
will always come to equilibrium in some finite amount of time, noting that 
the equilibrium position has its own uncertainty.

I hope this verbose answer gives you some things to think about.

Regards,
Dr. James Kranz