Comment on “Cooling by Heating: Refrigerator Powered by Photons”
Armen E. Allahverdyan1,4), Karen V. Hovhannisyan2) and Guenter Mahler3)
1)
arXiv:1302.4392v1 [cond-mat.stat-mech] 18 Feb 2013
Laboratoire de Physique Statistique et Systèmes Complexes,
ISMANS, 44 ave. Bartholdi, 72000 Le Mans, France
2)
ICFO-Institut de Ciencies Fotoniques, 08860 Castelldefels (Barcelona), Spain
3)
Universitat Stuttgart, 1. Institut fr Theoretische Physik, Pfaffenwaldring 57, 70550 Stuttgart, Germany
4)
Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia
Results obtained recently in Phys. Rev. Lett. 108, 120603 (2012) by Cleuren et al. apparently
contradict to the third law of thermodynamics. We discuss a scenario for resolving this contradiction,
and show that this scenario is pertinent for clarifying the general message of the third law.
1. A recent interesting letter by Cleuren et al. presents
a model for a refrigerator operating between two thermal
baths at temperatures Tc and Th , Tc < Th [1]. Among
other findings, Cleuren et al. report on a low-Tc regime,
where the heat Q̇c taken per time-unit from the lowtemperature bath scales as
Q̇c ∼ Tc .
(1)
If (1) is assumed to hold for all temperatures down to
Tc → 0, it would invalidate (for this model example) the
third law of thermodynamics [2]. This law states that the
rate of cooling of the low-temperature bath goes to zero.
The challenge of saving the third law amounts to showing
why (1) cannot hold down to very low temperatures Tc .
A proposal on why this might be the case for the model
studied in [1] was recently made by Levy et al. [2].
2. The purpose of the present note is to explain why
to our opinion the proposal by Levy et al. cannot be
accepted as a fundamental reason for saving the third
law, and then make our own attempt of arguing against
the validity of (1) at very low temperatures. We stress
that we do not make any claim on the invalidity of the
results by [1] within their model. It is the applicability
of the model for low temperatures that is questioned.
3. The contradiction between (1) and the third law is
deduced via routine thermodynamic considerations: (i)
the low-temperature thermal bath stays in overal equilibrium despite of its interaction with the refrigerator.
Hence it will respond to the refrigerator by lowering its
temperature. (ii) The rate at which its temperature Ṫc
is lowered can be evaluated within the linear response,
since the energy taken out due to refrigeration is much
smaller than the energy of the bath. Hence [2]
Ṫc =
Q̇c
,
c(Tc )
(2)
and taking into account that for Tc → 0 the constantvolume heat capacity c(Tc ) behaves at least as c(Tc ) ∼ Tc
for reasonable thermal baths (including the electron bath
studied in [1]), one concludes that Ṫc will be at least
constant for Tc → 0, which contradicts to the third law.
4. To save the third law, Levy et al. propose that the
Hamiltonian of the refrigerator employed in [1] is supplemented by another, physically well motivated term that
invalidates (1) for a low Tc [2]. This salvation of the third
law is not satisfactory for the following reason.
Any model of refrigerator must describe its specific
function. This description necessarily involves taking
“limits”, i.e. letting certain parameters in the respective Hamiltonian go to zero. We distinguish two types of
such asymptotic behavior:
1. “Circumstantial limits” strengthen the functional
charateristics of the model. Applying such limits is a
natural desire of building better devices. Indeed, good
devices do have rather special Hamiltonians. As the evolution of room refrigerators shows, even when their theoretical operating principles are clear, it still takes many
years and substantial engineering efforts to built good
devices, precisely because many unwanted terms in their
Hamiltonians are to be eliminated.
2. “Dysfunctional” limits would suppress the desired
function of the device (asymptotically).
Now Levy et al. based their arguments on an circumstantial limit as a potential reason of violating the third
law. It may be difficult to implement this limit in practice, but nothing in the analysis by Levy et al. shows
that the term they propose cannot in principle be made
as small as desired. This viewpoint on the salvation of
the third law would suggest that this law is not fundamental, but it holds due to imperfections present in the
Hamiltonian.
In contrast, we are going to argue that the violation of
the third law by Cleuren et al. relates to a dysfunctional
limit: if it is applied down to very low temperatures, the
basic functional characteristics of the model (its power of
refrigeration) will be harmed.
5. Now we explain why the weak-coupling masterequation-based refrigerator model by Cleuren et al. gets
limited at low Tc . An essential feature of such Markov
models is the detailed balance with respect to each thermal bath. Due to this, for Tc = Th = T (equal temperature baths) the refrigerator density matrix ρ has the
Gibbs form ρ ∝ e−H/T , where H is the refrigerator’s
Hamiltonian. For T → 0 this predicts that the refrigerator itself will be in its pure ground state. Such a
conclusion is impossible for a system permanently coupled to a thermal bath, provided that both the systembath interaction Hamiltonian and its commutator with
the full Hamiltonian stay finite (non-zero); see e.g. [3].
2
The model studied in [1] belongs to this class.
The usual way of understanding the low-temperature
limit of the Gibbs density matrix for an open system is
to assume that simultaneously with T → 0 the coupling
to the bath is made progressively smaller. However, for
the present situation this limiting process for justifying
the Gibbs density matrix down to low temperatures does
not work, since any refrigerator should have a finite coupling to the baths for ensuring a finite power of its operation. While the argument strictly speaking applies only
for Tc = Th = T , it should be clear that there are low-Tc
validity limits of the weak-coupling master equation also
for Tc < Th . Hence if the weak-coupling master equation
is forced to apply for all Tc → 0, its coupling to the lowtemperature bath should be made progressively weaker
nullifying Q̇c /Tc in (1).
6. The above argument on the inapplicability of the
usual Markov models at low temperatures suggests that
the analysis of the low-temperature refrigeration will certainly benefit from being carried out in a set-up, where
the refrigerator bath interaction is treated exactly, without any assumption on progressively weak refrigeratorbath interactions. Now one should and can ensure that
the power of refrigeration stays finite down to Tc = 0.
Such a model-dependent analysis has been carried out recently showing that albeits the regime (1) is reproduced
by sufficiently good devices at moderate and low values of
Tc , it is broken down for very low temperatures holding
the third law [4].
Our conclusion is that a proposition like (1) should
always be supplemented by demanding that the power
of refrigeration stays finite. Then presumably it cannot
hold down to very low temperatures, as the model studied
in [4] shows. If such a proposition could be shown to hold
down to Tc = 0 with only circumstantial limits involved,
it would constitute a a “real” violation of the third law.
[1] B. Cleuren, B. Rutten and C. Van den Broeck, Phys. Rev.
Lett. 108, 120603 (2012).
[2] A. Levy, R. Alicki and R. Kosloff, Comments on
cooling by heating: Refrigeration powered by Photons,
arXiv:1208.2600 [quant-ph].
[3] Th.M. Nieuwenhuizen and A.E. Allahverdyan, Phys. Rev.
E 66, 036102 (2002).
[4] A. E. Allahverdyan, K. Hovhannisyan and G. Mahler,
Phys. Rev. E 81, 051129 (2010).