I have started this thread off the back of the thread "Some Question" started by Saith.

Entropy is definately not a measureable quantity, not even for simple systems. Sure we construct a theoretical system of say fermions in a box, find all possible states, choose a particular configuration and CALCULATE its entropy, but we cannot get a measure of entropy without first measuring other properties of a system, eg temperature, pressure etc. It's not a technical incompetence, just like we are unable to measure the free energy of a system, even though a system will evolve to minimise its free energy.

Any thoughts

Hi ryans,

You just passed the "property of crisp" sign :D...

"Entropy is definately not a measureable quantity, not even for simple systems. Sure we construct a theoretical system of say fermions in a box, find all possible states, choose a particular configuration and CALCULATE its entropy, but we cannot get a measure of entropy without first measuring other properties of a system, eg temperature, pressure etc."

From a pure thermodynamics viewpoint, I don't think a direct measurement of entropy can be possible either; indirect measurements are surely possible. For example, you can use the differential equation:

dS = C dT / T

where dS is the (infinitesimal) change in entropy, C is the heat capacity, dependent on the type of process, i.e. if you are using constant volume or constant pressure, and dT is the (infinitesimal) change of the temperature T. If you know exactly how the heat capacity C depends on the temperature, you can perform the integration and, in terms of temperature changes, plot the entropy change (and I should add that chemists have been really busy measuring heat capacities up to order T⁴ dependence, and even higher I suppose).

This is a pure thermodynamical argument. From an equilibrium statistical mechanical point of view it becomes impossible I think: measuring the Boltzmann entropy is a real challenge :). But then again statistical mechanics reduces to ordinary thermodynamics in the thermodynamic (or hydrodynamic if you will) limit.

For non-equilibrium processes, changes in entropy can be related to the probability of observing specific behaviour of a system. In this way, one could perform an experiment over and over again, recording the frequencies of certain events to approximate the probabilities involved. The group I am working in has done work on that (yes, this is really an advertisement ;)), you can find more info at:

<A HREF="http://arxiv.org/abs/cond-mat/0202501">Time-reversal and entropy</A>
<A HREF="http://arxiv.org/abs/cond-mat/0211252">Quantum entropy production as a measure of irreversibility</A>

Oh by the way, you realized that I am talking about changes in entropy; performing an absolute measurement of entropy is ofcourse impossible, but the same goes for other quantities such as energy (you also need to fix a reference level).

"It's not a technical incompetence, just like we are unable to measure the free energy of a system, even though a system will evolve to minimise its free energy."

I disagree. You could use the so called Jarzynski equality (1997) to relate the work of a process with the change of free energy involved. The equation goes along the lines of:

exp( -<FONT FACE="symbol">b D</FONT>F ) = < exp( - <FONT FACE="symbol">b D</FONT>W ) >

where < . > stands for a proper average. You can solve this for the change in free entropy and measure it (well, indirectly perhaps, but I think most, if not all, measurements of energy are indirect). More info at:

C. Jarzynski; Phys. Rev. Lett. 78, (1997), 2690
C. Jarzynski; Phys. Rev. E 56, (1997), 5018

You can read the original paper at <A HREF="http://arxiv.org/abs/cond-mat/9610209">the Los Alamos pre-print</A> archive.

Perhaps you also want to shed some light on your new theory involving Poincare recurrences ? (cfr. other thread)

Bye!

Crisp

Crisp,

I had only a quick view through the second link you presented and could not believe my eyes. This is directly related to some work I've been doing, and in particular an idea I have been working on. The thing that suprised me is the definition of the time evolution operator that was outlined in the paper, as this is precisly the object which has been the centre of my attention. Are you one of the authors of the paper? I thought that what I had done was irrelevant, but I am now inspired to take it further.

I am not going to outline everything here, but the basic premise of reversibility goes back to Poincare' and Boltzmann, namely the apparent contradiction between Poincare cycles and Boltzmann's H thereom. I have tried to construct a model for entropy, which is a function of configuration of 6N dimension phase space (N is the number of particles in the system) for which a time evolution operator acts on to change the configuration such that the entropy of the new configuration is either equal to or greater than the previous configuration.

Hi ryans,

"The thing that suprised me is the definition of the time evolution operator that was outlined in the paper, as this is precisly the object which has been the centre of my attention. Are you one of the authors of the paper?

Hrm, we don't really define a time evolution in that paper, it is more a probability measure that is defined through some, unspecified unitary evolution. But the paper is quite compactly written (I am not the main author, merely cooperated there), so it requires some thinking about... You are ofcourse free to ask/mail questions ;).

"I thought that what I had done was irrelevant, but I am now inspired to take it further."

Great!

"I am not going to outline everything here, but the basic premise of reversibility goes back to Poincare' and Boltzmann, namely the apparent contradiction between Poincare cycles and Boltzmann's H thereom"

I would say it goes even further: on a macroscopic scale, Poincare cycles are neglegible (typically order 10^10^23 or similar for macroscopic objects). The main problem is, I think, the absense of time reversibility on a macroscopic scale, even though the underlying microscopic dynamics is time reversible. This problem is typically attributed to Boltzmann and the Stosszahlansatz he used in the derivation of the Boltzmann equation (for dilute gasses).

Poincare recurrencens offer an additional problem, but at much larger timescales. The breaking of time irreversibility occurs at more realistic timescales (i.e. seconds, days, weeks), so that is why I think it is more problematic :).

"I have tried to construct a model for entropy, which is a function of configuration of 6N dimension phase space (N is the number of particles in the system) for which a time evolution operator acts on to change the configuration such that the entropy of the new configuration is either equal to or greater than the previous configuration."

Hrmmmm.... There exists a very general result about this: if you have a consistent equation on macroscopic scale, then the (Boltzmann) entropy must increase, or at worst: stay equal. This result is discussed in the first paper I mentioned, <A HREF="http://arxiv.org/abs/cond-mat/0202501">time-reversal and entropy</A>. Lebowitz also discusses it in <A HREF="http://arxiv.org/abs/cond-mat/0304251">one of his recent papers</A>, claiming that it is new, even though the idea has been in textbooks here for 15 years.

Nevertheless, I would be interested to hear more from your ideas. I've just finished work on the Poincare recurrence vs macroscopic irreversibility problem (preprint in preparation), so perhaps we can exchange some ideas...

Bye!

Crisp