Radiation entropy bound from the second law of thermodynamics

It has been suggested heuristically by Unruh and Wald, and independently by Page, that among systems with given energy and volume, thermal radiation has the largest entropy. The suggestion leads to the corresponding universal bound on entropy of physical systems. Using a gedanken experiment we show that the bound follows from the second law of thermodynamics if the CPT symmetry is assumed and a certain general condition on matter holds. The experiment suggests that a wide class of Lorentz invariant local quantum field theories obeys a bound on the density of states.

It has been suggested heuristically by Unruh and Wald, and independently by Page, that among systems with given energy and volume, thermal radiation has the largest entropy. The suggestion leads to the corresponding universal bound on entropy of physical systems. Using a gedanken experiment we show that the bound follows from the second law of thermodynamics if the CPT symmetry is assumed and a certain general condition on matter holds. The experiment suggests that a wide class of Lorentz invariant local quantum field theories obeys a bound on the density of states. The demand that total entropy of a closed system never decreases imposes important constraints on the structure of microscopic theories. A particularly interesting consequence of the demand is the existence of universal upper bounds on entropy of physical systems. Such bounds can be inferred by considering gedanken experiments that involve a reference thermodynamical system interacting universally with systems from the considered class. The first example of such a reference system was provided by black holes. Using that any physical system interacts with a black hole, Bekenstein proposed a universal entropy bound where S is the system entropy, E is the energy and R is the system size [1]. The bound is based upon the demand that the total entropy never decreases in gedanken experiments that involve system absorption by a black hole. The derivation employs an extended version of the second law (the so-called generalized second law) that includes the contribution of the black holes in the total entropy. Despite some objections on the realizability of the considered gedanken experiment [2], no physical counterexamples to the bound were found since its proposition. Moreover, the bound was rederived by an alternative approach [3]. Here we wish to address another bound on entropy, related to a universal agent different from a black hole, namely the radiation. In their discussion of the Bekenstein bound, Unruh and Wald [2,4] (see also [5]) and Page [6] proposed heuristically that at given energy and system size, the entropy of radiation is the largest so that with coefficient of proportionality of order one. Following [7] we will refer to the above bound as the Page-Unruh-Wald (PUW) bound. The PUW bound expresses the expectation that at given E and R the state with the largest entropy is a gas (cf. [8]), and it takes into account that thermal radiation has the maximal concentration of about one particle per volume with size of the Compton wavelength. The bound is stronger than the Bekenstein bound and often the difference is large: the system size can be much larger than its Compton wavelength, R ≫ R Comp ≡ c/E. On the other hand, the bound (2) may fail in situations with strong gravity. For example, it fails for black holes that saturate (1). Still, the bound can be expected to hold quite generally and for its use it is important to understand what is needed for the bound to hold. Here we show that the second law of thermodynamics allows to shed light on the underlying assumptions.
We consider a weakly self-gravitating thermodynamic system with given energy E, volume V and entropy S. We assume that the CPT symmetry holds and the antimatter system obtained upon the CPT transformation of the original system is also physically admissible. Then, the antimatter partner is in the same thermodynamic state as the original system and it has the same energy E, volume V and entropy S. We now let the two systems, matter and antimatter one, interact. It can be assumed that before the moment of the interaction the two systems are enclosed in a somewhat larger box, so that there is a spacing both between both the systems and the box. The volume of the box can be chosen close to 2V . Before the interaction the entropy inside the box is given by (the entropy of the box is not essential in the following considerations): After the interaction the system and the "antisystem" annihilate giving rise to a certain product of the reaction. Designating the entropy of the product by S an (2E, 2V ), we find that the second law demand that the entropy of the whole system does not decrease as a result of the reaction produces the inequality In a wide class of situations the product of the annihilation reaction is the thermal radiation and we will have where we used that S rad (E, The scheme of the experiment in this case is shown in the Figure. Then the second law condition described by Eq. (4) produces the sought for Page-Unruh-Wald bound where the effective number of degrees of freedom (see [6,8]) and numerical constants are absorbed in the g−factor which is of the order of a few. It should be noted that we use the term "thermal radiation" in the spirit of the original works [2,4], where "radiation" includes matter with the rest energy smaller or comparable with the temperature, see also [8]. In particular, the g−factor in Eq. (6) is energy density dependent near temperatures corresponding to a pair creation threshold. We see that once the idea of the gedanken experiment is conceived, the demonstration of the bound is rather immediate. A possible objection to the consideration could concern the existence of an enclosure able to retain the products of the annihilation reaction. However, the assumption of the existence of such an enclosure can be avoided by noticing that we deal here with the volume (where the reaction occurs) versus the area (where the boundary is) problem. By making, if necessary, a large number of copies of the system-antisystem pairs and letting them interact in their common volume one can see that for the second law of thermodynamics to hold, the entropy should not decrease in the volume (the volume entropy grows faster with the number of copies than the contribution to the entropy from the boundary). It is the latter demand that is expressed by Eqs. (4) and (6).
Let us note that the bound in the form (6) may fail for containers where one dimension of size d is much smaller than the other two of size D. If d is much smaller than the thermal wavelength λ of the photons, while D ≫ λ, then we deal with effectively two-dimensional gas of photons where S ∝ (DE) 2/3 . The latter value exceeds (D 2 d) 1/4 E 3/4 following from (6) if d is sufficiently small. This and similar counterexamples to the bound can be avoided be rewriting the latter in its more fundamental form whereŜ rad (E, V ) is defined as the entropy of radiation that takes into account the given shape of the container.
In the form (7) the bound holds also for special shapes of container such as described above.
In the cases where gravity effects become important, the bound modifications are needed. In particular, the PUW bound has a different form for small system volumes, V R 4 g R −1 Comp , where R g ≡ GE/c 4 is the gravitational radius of the system [6]. Here the equilibrium state of thermal radiation includes a black hole that exchanges radiation with the system via the Hawking radiation process [9]. As a result, at these V one should use for S rad (E, V ) in Eq. (6) an expression that involves a sum of the entropies of the black hole and the remaining radiation, see [6].
As with any entropy bound there is a question of how precisely the bound works and how the situations where it could fail are avoided. An example of a challenge to entropy bounds is the zero mode argument proposed in [14]. It is argued that a zero mode possibly present in a field can accommodate arbitrarily large entropy at zero energy cost that would lead to a violation of an entropy bound. This argument against a bound was refuted in [15] where it was shown that in fact a zero mode contributes vanishing entropy to the system. A general discussion of how entropy bounds work can be found in [16].
The proposed gedanken experiment is informative about any system whatever the details of the product of annihilation reaction are. However, it seems most interesting where one can say that the product is radiation thus obtaining the PUW bound for the considered system. Then the physical reason for the bound is the expected instability of the matter-antimatter couple and the irreversibility of the resulting annihilation reaction. While radiation is expected to result from annihilation rather generally, it seems impossible to exclude on general grounds the possibility of unusual situations with other reaction results such as bound matter-antimatter states. In this case the bound in the form described by Eq. (4) will still work and one would have to calculate the entropy of the bound state.
Thus the main drawback of the proposed consideration is the absence of a condition to decide the product of the annihilation reaction. It is clear however that the product is radiation for most known real systems. Thus the present work demonstrates the bound for a wide class of systems occurring in nature. Moreover, it shows for the first time what may go wrong if the bound does not hold. By establishing that the assumption that annihilation leads to radiation implies the bound we get a new point of view on the bound that might eventually lead to significant progress in its understanding. Below we indicate two directions for future studies implied by the present work.
Our consideration shows that the bound can be expected to hold rather generally in theories that incorporate antimatter. Then the CPT theorem (see e. g. [10]) implies that a local Lorentz invariant quantum field theory (below LLIQFT) obeys the bound (4). Furthermore, in a general class of situations where the product of the annihilation reaction is given by the radiation, a LLIQFT must also obey the PUW bound. Using the Boltzmann formula for the entropy S(E, V ) = ln g(E, V )∆E, where g(E, V ) is the density of states and ∆E is the energy window, we conclude that density of states of a wide class of LLIQFT should grow with E not faster than exp[S rad (E, V )] (the factor in front of the exponent gives a subleading dependence on E). This sheds new light on the calculations of [11,12], aimed to demonstrate the validity of the holographic entropy bound for LLIQFT. The holographic bound S ≤ πc 3 R 2 / G was introduced by 't Hooft and Susskind [8,13] and it can be obtained from Eq. (6) by using R g < R. The above leads to the conjecture that under the assumptions of LLIQFT the PUW bound must hold. The study of this conjecture is a subject for further work.
Another direction that the present work suggests is the generalization of the PUW bound to include gravity. As mentioned in the introduction, the PUW bound does not hold in the situations with strong gravity where one should rather use the Bekenstein bound (1). Is it possible to derive a more general bound that would reproduce the two bounds in the limits of weak and strong gravity? Since the suggested gedanken experiment can be considered in situations where the self-gravity is not negligible, it is not unlikely that such a generalized bound exists. The study of this possibility is an important subject for further studies.
I am very grateful to J. D. Bekenstein for numerous helpful discussions and kind attitude. I also thank very much S. Nussinov for a very fruitful discussion. A. Casher is acknowledged for a useful comment and M. Vucelja for help in preparing the picture. This work was supported by DIP 0603215013 and BSF 0603215611 grants.