Cent. Eur. J. Phys. • 10(3) • 2012 • 708-714
DOI: 10.2478/s11534-012-0042-y
Central European Journal of Physics
Prior probabilities and thermal characteristics of heat
engines
Research Article
Preety Aneja1 , Ramandeep S. Johal1∗
1 Indian Institute of Science Education and Research (IISER) Mohali,
Knowledge City, Sector 81, Manauli P.O., Mohali-140306, Punjab, India.
Received 19 December 2011; accepted 12 March 2012
Abstract:
The thermal characteristics of a heat cycle are studied from a Bayesian approach. In this approach, we
assign a certain prior probability distribution to an uncertain parameter of the system. Based on that prior,
we study the expected behaviour of the system and it has been found that even in the absence of complete information, we obtain thermodynamic-like behaviour of the system. Two models of heat cycles, the
quantum Otto cycle and the classical Otto cycle are studied from this perspective. Various expressions for
thermal efficiences can be obtained with a generalised prior of the form Π(x) ∝ 1/x b . The predicted thermodynamic behaviour suggests a connection between prior information about the system and thermodynamic
features of the system.
PACS (2008): 02.50.Cw, 03.65.-w, 05.70.-a
Keywords:
prior information • quantum heat engines • Otto cycle • subjective probability
© Versita sp. z o.o.
1.
Introduction
The concept of information plays a very important role
in thermodynamics and has been explored from various
angles [1, 2]. In the present work, we explore a novel
connection between the notion of information and thermodynamic theory by asking the following question: How
can the available incomplete information about the system be utilized to predict its behaviour? The source of
this uncertainty is not due to some unknown intrinsic dynamics of the system or due to some environment induced
fluctuations of the control parameters. We do not assume
∗
E-mail: [email protected]
708
any such objective mechanism which could cause these
values to become uncertain. The issue is: can someone
who is ignorant of the exact values of certain system control parameters, make some reasonable estimates about its
behavior? In other words, how should one quantify one’s
subjective ignorance (or lack of information) to arrive at
good estimates?
In our opinion, this question can suitably be addressed
within a Bayesian framework, which nowadays is very
popular in Physics [3]. In the Bayesian approach [4], the
unknown parameter(s) of the system are taken as random variables with a certain prior probability distribution. Alternately stated, a prior is assigned to an unknown parameter. But assigning a unique prior for some
given prior information, has always been a crucial problem
in Bayesian analysis. The assignment might use certain
P. Aneja, R.S. Johal
principles such as invariance of the problem with respect
to some transformations [5–7]. Motivated by this principle of consistency, we propose arguments which lead to a
choice of the prior. To clarify, it is not the case that we are
completely ignorant of all aspects related to these parameters. There is some prior knowledge such as a possible
range of values, a relation between the different parameters following from Physics and so on. The proposed prior
is meant to take into account this known information. It
is not meant to imply that our prior is the unique prior
which quantifies this known information. Furthermore, in
the standard Bayesian approach, the inference process
provides a recipe to update the priors when further measurements are carried out on the system. If any such data
is not available, then we believe that the estimates have
to be based on the initial prior only and so the choice of
a “good” prior becomes more significant.
In this paper, we first argue in favour of a particular prior
to study the characteristics of heat engines. We then use
this prior to prove that estimated behavior of heat engines
can be remarkably close to the results of objective optimisation of these machines with respect to their external
control parameters. In particular, the different thermodynamic efficiences (such as the Curzon-Ahlborn (CA) efficiency [8] at maximum power when heat cycles work in
finite time and the other efficiences when the heat cycles
work with finite reservoirs (source/sink) [9–11], etc.) make
appearance in this approach which is based on prior probabilities. In the near-equilibrium regime, all efficiences
corresponding to the generalised prior exhibit a universal
form (ηcarnot /2) as also seen in other approaches [12, 13].
Thus our aim in this paper is to study the characteristics of heat engines from a Bayesian point of view. We
study Otto heat engine in two versions, one is the quantum and the other is the classical version. The cycle will
be reversible in the sense that it will perform work in infinite time. This approach was used in the quantum Otto
cycle with a two-level system to reproduce certain thermodynamic features thus pointing a connection between
thermodynamic behaviour and the concept of information
[14]. Here, the classical Otto cycle is also analysed from
the Bayesian perspective. In analogy with the quantum
Otto cycle, a prior distribution assigned to the intermediate temperature in the classical cycle can be employed to
estimate its thermodynamic features.
The paper is organized as follows. In Sec. 2, a model of
the quantum Otto cycle with a two-level system is discussed. In Sec. 3, the prior is chosen for the quantum
cycle, giving special emphasis to Jeffreys’ prior. Following
that, the use of a generalised prior and then the calculation of expected work for these priors is discussed. In
Sec. 4, classical reversible Otto cycle and the choice of
prior is discussed. Also a comparison is drawn between
the results of the quantum and classical versions of the
Otto cycle. Finally, in Sec. 5, the concluding remarks are
given.
2. Quantum Otto cycle with a twolevel system [15]
The quantum analog of a classical Otto cycle operating
between two heat baths at temperatures T1 and T2 , with
a two-level system with energy levels (0, a) and initial
state: a = a1 ; T = T1 , involves the following steps:
(i) The system is detached from the hot bath (T1 ) and
made to undergo the first quantum adiabatic process, during which a decreases from a1 to a2 , without causing any transition between the levels and
so the system continues to occupy its initial state.
The work done by the system is the change in mean
P
energy, W1 = i (Ei0 − Ei )pi .
(ii) The system with a = a2 is brought in thermal contact
with the cold bath (T = T2 ). The average heat reP
jected to the bath is Q2 = i Ei0 (p0i −pi ). Explicitly,
for the two-level system
1
1
−
1 + exp(a2 /T2 ) 1 + exp(a1 /T1 )
Q2 = a2
.
(1)
(iii) The system is now detached from the cold bath and
made to undergo a second quantum adiabatic process during which a2 → a1 . Work done on the
P
system in this stage is W2 = i (Ei − Ei0 )p0i .
(iv) Finally, the system is brought in thermal contact
with the hot bath again, from whence it recovers
its initial state. The heat absorbed from the hot
P
reservoir is Q1 = i Ei (pi − p0i ):
Q1 = a 1
1
1
−
1 + exp(a1 /T1 ) 1 + exp(a2 /T2 )
.
(2)
Total work done in one cycle is W = W1 + W2 =
Q1 + Q2
W(a1 , η) =
1
1
a1 η
−
> 0.
1 + exp (a1 /T1 ) 1 + exp (a1 (1 − η)/T2 )
(3)
with efficiency η ≡ W /Q1 = 1 − a2 /a1 .
709
Prior probabilities and thermal characteristics of heat engines
3. Bayesian approach and prior
probabilities
Consider that the reservoir temperatures (T1 , T2 ) and the
engine efficiency (η) are known a priori, but the value of
parameter a1 is uncertain. The question is: what can we
say about the expected values of the physical quantities
related to the performace of this engine? We approach
this question from a Bayesian viewpoint, where fundamentally, the notion of probability is regarded as epistemic i.e.
based on the state of knowledge of the observer and so
is subjective in nature. Thus an uncertain parameter is
assigned a prior probability distribution, which quantifies
our prior information about the likelihood for the parameter taking a certain value. For instance, if we have complete ignorance about its value, in a certain known range,
then it seems reasonable to assume a uniform distribution.
But this prior is not invariant under reparametrisations.
Jeffreys’ choice [16] Π(a1 ) ∝ 1/a1 for the prior, can be
argued in the present context as follows [17]. Having fixed
the value of efficiency η, there is essentially only one
energy scale say a1 , to be specified in the model, because
the other scale a2 is determined from the ratio a2 /a1 =
(1 − η). It is convenient to consider two observers A and
B. Let observer A treat a1 as the uncertain parameter
and denote the prior as Π(a1 ). The second observer B
chooses a2 as the uncertain parameter and denote the
corresponding prior as Π∗ (a2 ). The probabilities assigned
by A and B for a given choice of a1 and a2 , must satisfy
Π(a1 )da1 = Π∗ (a2 )da2 .
(4)
On the other hand, one should expect the same functional
form for the prior in both cases, which is saying essentially
that both A and B are in an identical state of knowledge,
implying Π ∼ Π∗ [18]. Thus relation (4) is rewritten as
Π(a1 ) = (1 − η) Π(a1 (1 − η)),
(5)
This suggests that the prior for volume should be the same
as for length, or in other words, the prior should be invariant under the change in dimension of space. This implies
the choice, Π(V )dV = Π(L)dL ∝ dL/L. Then the analogy
between V and a1 suggests the prior for the latter to be
Π(a1 ) ∝ 1/a1 .
3.1.
Expected work with Jeffreys’ prior
The normalised Jefferys’ prior is given by:
−1
1
amax
,
Π(a1 ) = ln
amin
a1
where amin and amax are the assumed minimal and maximal
energy splitting achievable for the two-level system.
Expected work over a given prior is defined by:
Z
W (η) =
710
W(a1 , η) Π(a1 )da1 ,
(7)
and with Jeffreys’ choice, we obtain
−1 amax
1 + eamax (1−η)/T2
T2
W = ln
ln
η
amin
(1 − η)
1 + eamin (1−η)/T2
amax /T1
1+e
.
(8)
−T1 ln
1 + eamin /T1
The expected work W vanishes in two cases: when η = 0
and η = ηcarnot . In between these values of η, the expected
work exhibits a maximum.
In the asymptotic limit, amin << T2 and amax >> T1 , the
work is approximated as:
ln 2
η T1 −
W ≈
ln
a functional equation whose solution is given by Π(a1 ) ∝
1/a1 .
On a more heuristic level, one may reason as follows. First
note that the parameter a1 can be regarded as the thermodynamic analog of volume V [19]. Thus it is reasonable
to assume that the prior will have the same form for either
of them. To choose a prior for the volume, note that volume
scales with length (L) as V ∼ Ld , where d is the dimension
of the space. On the other hand, a change in volume can be
achieved by changing only the length, while keeping the
area of cross-section A fixed, as for instance by moving the
piston in a cylinder. More precisely, we write dV = A dL.
(6)
amax
amin
T2
(1 − η)
.
(9)
In this case, maximum work is obtained at the efficiency
well-known as the Curzon-Ahlborn (CA) value:
s
η∗ = 1 −
T2
,
T1
(10)
which is a basic result in endoreversible thermodynamics
where it is the efficiency at maximum power.
P. Aneja, R.S. Johal
1.0
3
0.8
Η*
0.6
0.4
0.2
P
0.0
0.0
0.2
0.4
0.6
0.8
2
1.0
4
Θ
Figure 1.
η∗ vs. θ. The dashed curve is for Jeffreys’ prior (γ = 1)
and solid is for uniform prior (γ = 0)
1
.
V2
3.2.
Generalised prior
Figure 2.
Now we will choose a more general prior. It includes
Jeffereys’ prior and the uniform prior, as special cases.
Consider Π(a1 ) = Na1 −γ , defined in the range [0, amax ],
where N = (1 − γ)/amax 1−γ and γ < 1. For the asymptotic
limit, the efficiency at optimal expected work is given by
[14]:
(1 − η∗ )3−γ − (1 − γ)θ 2−γ η∗ − θ 2−γ = 0,
θ 4/3
= 1+ q
3 1+ 1+
r
θ2
27
1/3 −θ
2/3
1+
θ2
1+
27
V1
Pressure-Volume diagram of a reversible Classical Otto
cycle.
idealized cycle consists of two reversible adiabatic segments 1 → 2 and 3 → 4 and two reversible constant volume parts 2 → 3 and 4 → 1. The maximum and minimum
temperatures correspond to reservoir temperatures T3 and
T1 . Along the heating and cooling segments, the temperature varies from T2 to T3 and from T4 to T1 , respectively.
The heat transfers to the fluid, which are accomplished via
the paths 2 → 3 and 4 → 1, are:
(11)
where θ = T2 /T1 . Now as γ → 1, the above equation
yields the CA value as the solution. The ’optimal’ efficiency with uniform prior is obtained with γ = 0,
∗
ηγ=0
V
Qin = Cv (T3 − T2 )
Qout = Cv (T4 − T1 ).
!1/3
.
Because the cycle is reversible, the fluid’s entropy change
per cycle is zero:
(12)
4 S = Cv ln (T3 /T2 ) + Cv ln (T1 /T4 ) = 0.
4. Bayesian approach in classical
heat cycle
4.1.
Classical Otto cycle [20]
Classical Otto cycle is a reversible model of an internal
combustion engine operating between two extreme temperatures. This cycle consists of four branches, two of
which are adiabatic and two are isochoric (constant volume). Assuming the working fluid to be an ideal gas with
constant heat capacity, the cycle is as shown below: The
(13)
This gives
T4 =
T1 T3
.
T2
(14)
Since T1 and T3 are the fixed temperatures, the adjustment can only be made in T2 and T4 . But these are not
independent as shown in the relation above.
The work done per cycle, W = Qin − Qout , is given by:
W = Cv (T3 + T1 − T4 − T2 ).
(15)
711
Prior probabilities and thermal characteristics of heat engines
Similarly efficiency, η = W /Qin , using (14) can be written
as:
η =1−
T1
.
T2
T2∗ = T4∗ = (T1 T3 ) 2 .
1
Using the above values of T2∗ and T4∗ in (15), we obtain
the expression for maximum work:
Wmax = Cv T3 [1 −
P(T4 )dT4 = P(T2 )dT2 .
(16)
For fixed values of T1 and T3 , we will see for which values
of T2 and T4 W is maximised. Differentiate (15) w.r.t T2
and T4 , and we get [21]:
√
form for their priors. Then, equivalent to the quantum case
(Eq. (4)), we require that
θ]2 ,
θ=
T1
.
T3
(17)
√
Efficiency at maximum work is given by η∗ = 1− θ. This
is CA-efficiency.
In general, we can write
W
η(ηc − η)
,
=
T3 C v
(1 − η)
(18)
(20)
Then by using the fact T4 = T1 T3 /T2 , we simply obtain
P(T2 ) ∝ 1/T2 .
Of interest here is the expected value of efficiency which is
RT
defined as hηi = T13 ηπ(T2 )dT2 . Then for Jeffreys’ choice,
we have
(1 − θ)
,
(21)
hηi = 1 +
ln θ
whereas with the uniform prior, it is given by
hηi = 1 +
θ(1 − θ)
.
ln θ
(22)
As a side remark, we observe the effect of using a generalised power law prior for the classical case too. As for the
quantum case, this prior serves to incorporate the uniform
prior and the Jeffreys’ prior in a unified way. Thus a power
law probability distribution for the unknown temperature
T2 is assigned:
Π(T2 ) = R T 3
1/T2b
1/T2b dT2
1
1−b
.
1 − θ 1−b T2b
T 1
where ηc = 1 − T1 /T3 is the Carnot efficiency. This
expresses the relation between W and η. For a given ηc ,
lim [W ] = lim [W ] = 0.
η→0
η→ηc
(19)
Thus the Otto cycle gives zero work output at both its
maximum and minimum efficiencies.
4.2.
Prior in classical Otto cycle
To consider a situation where we have partial or incomplete information about the system, we have to identify
the thermodynamic control parameters of the problem. Although V1 and V2 correspond respectively to the parameters a1 and a2 of the quantum case, we note that the
work expression for the classical case, can be written in
two equivalent ways. In Eq. (18), work is a function of
efficiency η only. Alternately, work can be expressed in
terms of intermediate temperature which can be either T2
or T4 , due to Eq. (14). Thus either we assign prior for η,
or alternately to T2 (or T4 ).
Again, we consider two observers A and B who assign prior
for T2 and T4 respectively. Each temperature is defined
in the range [T1 , T3 ]. As the state of knowledge for A and
B is the same, they can be assigned the same functional
712
Π(T2 ) =
(23)
where b = 0 (uniform) and b = 1 (Jeffreys) represent
special cases.
The expected efficiency corresponding to the power-law
prior is given by:
hηi = 1 +
(1 − b) (θ − θ 1−b )
.
b (1 − θ 1−b )
(24)
Table 1 shows the expressions for hηi using different
values of b. It is seen that hηi is a monotonic function
of b, and interpolates between Carnot efficiency and
zero value as b ranges from −∞ to ∞. For θ close to
equilibrium (θ ≈ 1), hηi can be approximated as:
hηi =
ηc
1
+
(2 − b)ηc 2 + O(ηc 3 )
2
12
(25)
Finally, we compare the classical analysis with our quantum model. Note that because of the monotonic relation
between efficiency and T2 , one can also choose η as the
unknown variable in the classical case. The corresponding situation in the quantum Otto cycle would be that
one parameter (say a1 ) is fixed and we assign prior to
P. Aneja, R.S. Johal
the quantum result is independent of which of the parameters a1 or a2 is held fixed. This feature is true only for
Jeffreys’ choice of prior (b = 1).
Table 1.
b
hηi
Comments
-∞
1−θ
-1
1−θ
1+θ
0
1+
Carnot efficiency.
θ ln θ
1−θ
1
2
1−
1
1+
√
Finite heat source
and infinite sink
[9, 10].
θ
CA efficiency [8].
ln θ
1−θ
Finite heat sink
and
infinite
source [9].
ηc /2.
1−θ
2
2
∞
0
Zero efficiency.
1.0
XΗ\
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Θ
Figure 3.
5.
Conclusions
ηc /(2 − ηc ) [22].
hηi vs. θ for different b’s. hηi is a monotonic decreasing
function of b as shown in Table 1. Here, the upper straight
line is for Carnot value ηc and then we have plotted for
b = −1, 0, 1/2, 1, 2, from top to bottom. Near Equilibrium
(θ ≈ 1), hηi exhibits a universal form independent of b and
given by hηi ≈ ηc /2.
the other (a2 ). Then, because the quantum efficiency is
η = 1 − a2 /a1 , it is equivalent to assigning a prior for the
efficiency.
Now for a given value of a1 , the values of a2 which lead
to an engine operation are in the range [a1 θ, a1 ]. Thus
the (conditional) Jeffreys’s prior for a2 will be
π(a2 |a1 )da2 =
1 da2
.
ln(1/θ) a2
Concluding, we have discussed an intriguing connection
between prior probabilities which quantify prior information and thermal efficiencies of heat engines. These probabilities reflect the degree of belief of an observer who
only has access to incomplete prior information about the
system. In the present context, this uncertainty is introduced through a lack of knowledge about the exact values
of the macroscopic control parameters. Consistency demands that different observers who have access to similar
prior information should assign similar priors. We have
considered two observers, each is uncertain about an energy scale of the system. Further input about the system
or the physical process is in the form of a constraint like a
given value of the efficiency or the relation between intermediate temperatures following a cyclic process. Thus we
arrive at Jeffreys prior for the uncertain internal energy
scale of the working medium, in both the quantum as well
as the classical version of the Otto heat cycle. The well
known Curzon-Ahlborn efficiency follows as the efficiency
at the maximum of the work averaged over the prior distribution for the quantum model. We also observe that using
slightly different priors leads to other efficiences which are
close to the CA value. For the classical model, power-law
type of priors upon averaging suggest many well known
expressions for efficiency which have been previously obtained from very different approaches such as finite time
thermodynamics, finite source/sink set ups for engines and
so on. Also, for the particular quantum model, the result of
optimising work per cycle over the internal scales, yields
the result [23] that efficiency at optimum work is bounded
from below by the CA value. For optimising near to the
equilibrium, the efficiency can be closely approximated by
this value (which is, in this limit, half of the carnot value).
Although we have argued in favour of the π(a) ∝ 1/a kind
of prior, the use of the power-law form is not very clear.
What the present analysis still reveals is that there is a
non-trivial connection between expected optimum behavior based on the notion of subjective ignorance for the
concerned physical process and the objective optimisation
using concrete realisation for all control parameters.
(26)
Ra
Then we have hηi = a11θ η π(a2 |a1 )da2 . Interestingly,
the quantum model yields the same value of expected efficiency as in the classical case (see Eq. (21)). Furthermore,
This resemblence is startling, though we cannot insist that
there is a unique prior for the considered uncertainty of
control parameters. But what is expected of a prior is to
provide a good enough estimate, and in the present context, proximity to CA value is that result. Starting from
713
Prior probabilities and thermal characteristics of heat engines
this prior, the Bayesian program of updating the prior can
be implemented if some observational data is provided
further. In conclusion, we indicate that apart from thermodynamic origins, the various thermal efficiencies may
have an epistemic origin also, linking them to our state of
knowledge. This suggests, in our opinion, an outstanding
issue as to whether a deeper relation exists between the
notion of subjective information, as quantified in Bayesian
statistics, and the theory of thermodynamics. In future
work, we hope to cast further light on this highly interesting behaviour.
6.
Acknowledgements
RSJ thanks Prof. Sumiyoshi Abe and Prof. Georgio Kaniadakis for an opportunity to present his research efforts at
Sigma-Phi Conferene 2011, Larnaca, Cyprus. The authors
also acknowledge constructive comments from the anonymous referee which have helped in a clearer presentation
of the ideas in this paper. RSJ is supported by Department of Science and Technology, India under the research
project No. SR/S2/CMP-0047/2010(G). Preety Aneja is
thankful to UGC, New Delhi, India for Junior Research
Fellowship.
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