Thermodynamic laws for any observable


  • Are there additional laws of thermodynamics at the microscopic scale?

  • Can we do better than the second law at the microscopic scale when the dynamics is typically highly irreversible (out of equilibrium)?

  • Is it possible that any observable (not just energy related) is limited by a second-law-like constraint?

The laws of thermodynamics are truly fantastic. In particular, we understand now that careful statements of the second law hold all the way from the macroscopic scale to the microscopic scale. That said, there is a reason why the laws of thermodynamics are macroscopically so successful, while in the microscopic world their utility has not been established yet. First, macrocopic systems are often in local equilibrium. For example, when a piece of metal is connected to cold environment on one side, and to hot environment on the other side, every point in locally in thermal equilibrium but the temperature varies in space and time. Although formally the whole setup is not equilibrium the local equilibrium approximation greatly simplifies the dynamics. Unfortunately, local equilibrium is often no true in microscopic scale.
The second law holds even far from equilibrium, so what is the problem? The second law is an inequality, and as such it often overestimates the actual results. The closer the dynamics is to being reversible (i.e. close to equilibrium) the more accurate the prediction of the second law. As mentioned before in the microscopic world the dynamics is often out of equilibrium, so the second law is not that useful (for example, it significantly overestimates the actual values of heat flows) even though it holds. There is room than for finding better forms of the second law in the microscopic out of equilibrium regime. 

A second problem with the second law is that involve very specific quantities heat, entropy and temperature. The heat is the change in the average energy of the bath and the entropy related to the information content in the system. In the microscopic world this is useful since change in energy and entropy can be used to determine the new state of the system according to the equilibrium equation of state. However, in the macroscopic world we can now measure very fine details, for example the population of a specific energy level of a specific particle. Such quantities do not appear in the second law and it not clear if and to what extent thermodynamics constrains them. Fine-grained quantities and their thermodynamic constraints are directly related to the performance of exotic heat machines (X machines).

However, regardless of machines and tasks it is also of interest to study the thermodynamics limitations of higher moments of the energy as they contain valuable information. For example, the second moment of the energy of the system provide information on the spread of the energy distribution.  The second moment of the energy of the system-environment contains information on the system-environment correlation. In the second law, only the average energy appears (heat, work). That is, the first moment of the energy. Thus a pending question is can higher moments of the energy be manipulated without any limitation, or perhaps thermodynamics limits how much the can change. Alternatively stated, does thermodynamics relates changes in higher moments of one element (e.g. the system) to changes in some other elements (e.g. one of the environments).

In [1] I used the local approach to find, in some scenarios, constraints that have the form of the second law the related changes in higher moments of the energy to alternative measures of information. The Shannon/von Neuman entropies are extensive information measures. However, higher moments of the energy are not extensive, so it makes sense that information in new energy-information relations will be non-extensive as well. In [1] these relations were used to gain insight into blind spots which are not addressed by the standard second law. These new energy-information relations are fully compatible with quantum dynamics. Unfortunately, the regime of validity of these inequalities has not been fully mapped yet - only specific classes of scenarios where they hold has been identified.
To the best of my knowledge, this is also the first use of the Bregman divergence in physics. This is a powerful mathematical tool that is well suited for microscopic thermodynamic investigation.  

In [2] I undertook a completely new approach to the second. I introduced the concepts of global passivity and found that it broader and deeper than the second law. First, it fully retrieves the standard second law if we restrict it to the regime of validity of the second law (e.g. all elements must be initially uncorrelated). Second, it yields a more general form that holds even in the presence of initial quantum correlation between the elements. Third, it produces a family of inequalities on higher moments of the energy where the second law is just one of them. In this paper I thought of a new use for inequalities on higher order moments. In the same way Maxwell's demon violates the second law if it is taken into account, these new relations also break down in the presence of feedback (demon). There are however demons, called lazy demons, that cannot be detected by an apparent violation of the second law but can be detected by these new global passivity inequalities. This give rise to the interesting question if all feedback operations even the faintest ones can, in principle, be detected by some thermodynamic principle.
A more practical variant of this question concerns hidden heat leak detection imagine a quantum computer or some other small system that is supposed to be isolated from the environment. Is it possible to detect tiny heat leaks in cases where the energy balance is impossible to track since the system is aperiodically driven? Once again, we have demonstrated in [2] that the higher order passivity inequality can detect smaller heat leak compared to the second law.


Both of these approaches ([1]&[2]) are explained in [3] along with a broader view of the second law in microscopic setups.

In my research, I plan to extend the finding in [1,2] and overcome the deficiencies of the current results.



  1. R. Uzdin “Additional energy-information relations in thermodynamics of small systems” Physical Review E, 96, 032128 (2017)

  2. R. Uzdin and S. Rahav “Global Passivity in Microscopic Thermodynamics” Physical Review X, 8, 021064.(2018)

  3. R. Uzdin, "The second law and beyond in microscopic quantum setups", arXiv:1805.02065 (2018)