While $$\delta Q=T\ dS$$ allows for obtaining the entropy change of an isolated system in equilibrium by measuring the heat exchange with the environment, I was wondering whether there are ways to obtain an entropy field distribution, potentially also in non-equilibrium systems?
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asked Aug 2, 2014 at 10:01
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$\begingroup$ The question looks rather vague, maybe you could elaborate a bit more. Entropy production in non-equilibrium systems has been studied in different circumstances, one of them is via the study of entropy flux using a Fokker-Planck type of equation, an exemplary work can be found here: scielo.br/… $\endgroup$– EllieCommented Aug 2, 2014 at 11:12
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1$\begingroup$ @Phonon that's an interesting read there, thanks! concerning the vagueness, my question boils down to whether a device similar to a voltmeter or thermometer exists for entropy. $\endgroup$– Tobias KienzlerCommented Aug 2, 2014 at 11:15
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$\begingroup$ Well in that case then the answer is no, you can only measure entropy by studying the change of its dependancies on extensive/intensive variables that influence it (as they are different in different systems). To make more contrast, take the example of liquid crystals, where one has to define an appropriate order parameter to study and to be able to quantify the entropy change of the system. (e.g. nematic order parameter, density, etc.) $\endgroup$– EllieCommented Aug 2, 2014 at 11:25
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2 Answers
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Regarding your new formulated question "my question boils down to whether a device similar to a voltmeter or thermometer exists for entropy":
Well in that case then the answer is no, you can only measure entropy by studying the change of its dependencies on extensive/intensive variables that influence it (as they are different in different systems). To make more contrast, take the example of liquid crystals, where one has to define an appropriate order parameter to study and to be able to quantify the entropy change of the system. (e.g. nematic order parameter, density, etc.)
EDIT: An edit to further elaborate on the matter at hand, since from the comments it is clear that some folks may still not be convinced.
Quantities like entropy or chemical potential do not come about as directly accessible in a lab. What we call "measurable" typically comes with a mechanical response like pressure, bulk quantities or thermal ones like temperature and heat flow (i.e. studying $T$ changes in coupled systems). Hence we call $T$, $P$, $V$, $\Delta H_{latent}$ etc. measurable thermodynamic variables. Additionally there are also response functions that are measurable, as they correspond to a change in a system parameter in response to a perturbation, e.g. the expansion coefficient $\alpha_{p} \propto \delta V / \delta T$ at constant $P$, or heat capacities $C_V$, $C_P$, of course this is barely the full list (considering also combined response functions).
Now having mentioned a handful of measurable variables, next step is to see how these make the experimentally inaccessible quantities measurble. This is done using the famous Maxwell relations, an example:
Starting from an expression containing non-measurable quantities:
(Using the expression of the expansion coefficient)
$$\begin{align}
\left(\frac{\delta S}{\delta p}\right)_{N,T} &= - \left(\frac{\delta V}{\delta T}\right)_{N,P} \\
\left(\frac{\delta S}{\delta p}\right)_{N,T} &= - V\alpha_{P}
\end{align}$$
Which simply gives us the entropy dependance on pressure (at constant $T$) is related to thermal expansion of a system. One can keep using Maxwell relations to further simplify the expressions and ultimately reach relations only containing measurable ones.
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3$\begingroup$ In another line, entropy is a physical concept, originated in thermodynamics, but ultimately interpreted from Boltzman relation as a measure of the number of microstates of available to the system, thus related generally to the geometrical availability of configurations of the systems. This means that you could in general estimate from first principles when knowing the microscopic details of your system, but you cannot measure it directly. This is similar to the case of energy meaning that you find it by its dependence to other measurable variables, but it cannot be measured directly. $\endgroup$– rmhleoCommented Aug 2, 2014 at 13:36
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3$\begingroup$ @Phonon: But if held at that standard, wouldn't a classic thermometer also not measure temperature, since the measured quantity ultimately is the volume of the mercury in it? $\endgroup$– celtschkCommented Aug 2, 2014 at 18:44
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1$\begingroup$ @TobiasKienzler no problem, hope it clarified some things. But "how to generally measure entropy" does not have a unique answer as it directly depends on the system under question. In the last example I showed, it clearly demonstrates how in that specific example, change of entropy with respect to change of pressure is measured: $-V\alpha_P$ is the answer for that case, meaning if we know the volume of the system, we just need to define the thermal expansion coefficient of our system, i.e. $\partial V/\partial T.$ There's no entropy-meter that would work on any system, unlike a thermo-meter. $\endgroup$– EllieCommented Aug 7, 2014 at 9:47
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3$\begingroup$ @TobiasKienzler What you could do, would be to let your system evolve towards equilibrium after a small perturbation (of any kind), and see how the equilibrium differs from the last one, that would give you an idea of the entropy. For example when you wash your clothes with hot water, afterward they've shrunk down a bit, that is because your clothes are made of polymers that have been heated, stretched into a low entropy configuration and then super-cooled, so once you heat them a bit, they get the chance to re-conform themselves and evolve towards a higher-entropic configuration. $\endgroup$– EllieCommented Aug 7, 2014 at 9:59
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3$\begingroup$ so by looking at the difference of the size of your clothes afterwards, you would only get a qualitative sense of the entropy, conformational here, and bear in mind that each system has multiple entropic terms, configurational (atomic level), orientational (molecular level), homogeneity, etc. So for a entropy-meter to exist it would have to able to perturb your system at atomic, molecular and larger scales, which is impossible. Long story short, entropy is a state function that may depend on many many different quantities, so for it to be measured, you would need everything-meter :) $\endgroup$– EllieCommented Aug 7, 2014 at 10:04
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$$s=k_B \ln(v_2/v_1)^N$$ so if you are able to measure all the quantities present in this equation, you will be able to measure entropy. Here, $k_B$ is the Boltzmann constant and $N$ the number of molecules in the system.
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