Necessary and Sufficient Condition for Stability of Generalized Expectation Value

Abstract

A class of generalized definitions of expectation value is often employed in nonequilibrium statistical mechanics for complex systems. Here, the necessary and sufficient condition is presented for such a class to be stable under small deformations of a given arbitrary probability distribution.

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Given a probability distribution {𝑝_𝑖}_{𝑖=1,2,…,𝑊}, that is, 0≤𝑝_𝑖≤1 (𝑖=1,2,…,𝑊) and ∑^𝑊_𝑖=1𝑝_𝑖=1, the ordinary expectation value of a quantity 𝑄 of a system under consideration is defined by ∑^𝑊_𝑖=1𝑝_𝑖𝑄_𝑖, where 𝑊 is the total number of accessible states and is enormously large in statistical mechanics, typically being 2¹⁰²³. In the field of generalized statistical mechanics for complex systems, on the other hand, discussions are often made about altering this definition. Among others, the so-called “escort average’’ is widely employed in the field of generalized statistical mechanics [1–3]. It is defined as follows: ⟨𝑄⟩_𝜙𝑝=^𝑊_𝑖=1𝑃_𝑖^(𝜙)𝑄_𝑖, (1) where 𝑃_𝑖^(𝜙) stands for the escort probability distribution [4] given by 𝑃_𝑖^(𝜙)=𝜙𝑝_𝑖∑^𝑊_𝑗=1𝜙𝑝_𝑗, (2) with a nonnegative function 𝜙. In the special case when 𝜙(𝑥)=𝑥, ⟨𝑄⟩_𝜙 is reduced to the ordinary expectation value mentioned above.

Consider measurements of a certain quantity of a system to obtain information about the probability distribution. Repeated measurements should be performed on the system, which is identically prepared each time. Suppose that two probability distributions, {𝑝_𝑖}_{𝑖=1,2,…,𝑊} and {𝑝^′_𝑖}_{𝑖=1,2,…,𝑊}, are obtained through the measurements. They may slightly be different from each other, in general. If such measurements make sense, then the expectation values, ⟨𝑄⟩[𝑝] and ⟨𝑄⟩[𝑝^′], calculated from these two distributions should also be close to each other. This condition, which implies “experimental robustness,” is represented as follows.

Definition (stability). An expectation value ⟨𝑄⟩[𝑝] is said to be stable, if the following predicate holds for any pair of probability distributions, {𝑝_𝑖}_{𝑖=1,2,…,𝑊} and {𝑝^′_𝑖}_{𝑖=1,2,…,𝑊}: (∀𝜀>0)(∃𝛿>0)(∀𝑊)𝑝−𝑝^′₁<𝛿⟹⟨𝑄⟩𝑝−⟨𝑄⟩𝑝^′<𝜀. (3)

Here, ‖𝑝−𝑝^′‖₁=∑^𝑊_𝑖=1|𝑝_𝑖−𝑝^′_𝑖| is the 𝑙¹-norm describing the distance between these two probability distributions. One might consider norms of other kinds, but what is physically relevant to discrete systems is the present 𝑙¹-norm [5]. Equation (3) is analogous to Lesche’s stability condition on entropic functionals [5], which has recently been revisited in the literature [6–11] (note that the discussion in [8] is corrected in [9]). This concept of stability is actually equivalent to that of uniform continuity.

In recent papers [12, 13], it has been shown that the generalized expectation value in (1) with a specific class,𝜙(𝑥)=𝑥^𝑞(𝑞>0), (the associated expectation value being termed the 𝑞-expectation value), is not stable unless 𝑞=1. This result needs the 𝑞-expectation-value formalism of nonextensive statistical mechanics [1, 2] be reconsidered. In addition, the result is supported by Boltzmann-like kinetic theory in an independent manner [14].

Here, it seems appropriate to make some comments on the latest situation of the problems concerning stabilities of entropic functionals and generalized expectation values. The authors of [15, 16] have presented discussions which aim to rescue the 𝑞-expectation values from the difficulties of their instability pointed out in [12]. Those authors insist that the 𝑞-expectation values can be stable in both the finite-𝑊 and continuous cases. Such possibilities are, however, fully refuted by the work in [13] both physically and mathematically, and the controversy seems to have been terminated with that work. The case of the continuous variables has further been carefully examined in a recent paper [17], where the so-called Tsallis 𝑞-entropies [1, 2] do not have the continuous limit in consistency with the physical principles such as the thermodynamic laws (see also [18, 19]). These controversies have led the researchers to give up the traditional form of nonextensive statistical mechanics based on the 𝑞-entropies and 𝑞-expectation values and to examine other entropic functionals combined with the ordinary definition of expectation values [20] (see also [21, 22]). Thus, it seems that nonextensive statistical mechanics has to be fully reexamined, theoretically.

In this paper, we present the necessary and sufficient condition for ⟨𝑄⟩_𝜙[𝑝] in (1) to be stable.

Our main result is as follows.

Theorem. Let 𝜙 be nonnegative and continuous on [0,1], differentiable on (0,1), and satisfy the condition that 𝜙(x)=0⇔x=0. And, let Q={Q_i}_i=1,2,…,W be a random variable. Then, ⟨Q⟩_𝜙[p] in (1) is stable, if and only if lim_x→+0𝜙(x)/x∈(0,∞).

Proof. First, assume that lim_𝑥→+0𝜙(𝑥)/𝑥=𝑎>0. Then, there exists 𝛿₁>0 such that 𝑎𝑎−2<𝜙(𝑥)𝑥𝑎<𝑎+2∀𝑥∈0,𝛿₁. (4)𝜙(𝑥)/𝑥 does not vanish because of the condition 𝜙(𝑥)=0⇔𝑥=0. Therefore, there exists 𝑏>0 such that 𝜙(𝑥)𝑥≥𝑏∀𝑥∈𝛿₁,1. (5) Putting 𝑐=min{𝑎/2,𝑏} we have 𝑐𝑥≤𝜙(𝑥)(∀𝑥∈[0,1]). (6) Consequently, for an arbitrarily large 𝑊 and an arbitrary probability distribution {𝑝_𝑖}_{𝑖=1,2,…,𝑊}, we obtain 1∑^𝑊_𝑖=1𝜙𝑝_𝑖≤𝑐. (7) From the mean value theorem, it follows that 𝜙𝑝_𝑖−𝜙𝑝^′_𝑖≤𝑝_𝑖−𝑝^′_𝑖⋅sup_𝑥∈(0,1)𝜙^′(𝑥), (8) where 𝜙^′(𝑥) is the derivative of 𝜙(𝑥) with respect to 𝑥. For 𝜀>0, we put 𝛿=inf𝛿₁,𝑐𝜀2𝑄_max⋅sup_𝑥∈(0,1)𝜙^′(𝑥), (9) where 𝑄_max=max{𝑄_𝑖}_{𝑖=1,2,…,𝑊}. Now, for ‖𝑝−𝑝^′‖₁<𝛿, we have ⟨𝑄⟩_𝜙𝑝−⟨𝑄⟩_𝜙𝑝^′=1∑^𝑊_𝑖=1𝜙𝑝_𝑖∑^𝑊_𝑗=1𝜙𝑝^′_𝑗^𝑊_𝑖=1𝑄_𝑖𝜙𝑝_𝑖^𝑊_𝑗=1𝜙𝑝^′_𝑗−𝜙𝑝^′_𝑖^𝑊_𝑗=1𝜙𝑝_𝑗≤1∑^𝑊_𝑖=1𝜙𝑝_𝑖∑^𝑊_𝑗=1𝜙𝑝^′_𝑗×^𝑊_𝑖=1𝑄_𝑖𝜙𝑝_𝑖−𝜙𝑝^′_𝑖^𝑊_𝑗=1𝜙𝑝^′_𝑗+𝜙𝑝^′_𝑖^𝑊_𝑗=1𝜙𝑝_𝑗−^𝑊_𝑗=1𝜙𝑝^′_𝑗≤1∑^𝑊_𝑗=1𝜙𝑝_𝑗^𝑊_𝑖=1𝑄_𝑖𝜙𝑝_𝑖−𝜙𝑝^′_𝑖+∑^𝑊_𝑗=1𝜙𝑝_𝑗−𝜙𝑝^′_𝑗∑^𝑊_𝑖=1𝜙𝑝_𝑖∑^𝑊_𝑗=1𝜙𝑝^′_𝑗^𝑊_𝑖=1𝑄_𝑖𝜙𝑝^′_𝑖≤2𝑄_max∑^𝑊_𝑗=1𝜙𝑝_𝑗^𝑊_𝑖=1𝜙𝑝_𝑖−𝜙𝑝^′_𝑖≤2𝑄_max∑^𝑊_𝑗=1𝜙𝑝_𝑗𝑝−𝑝′₁⋅sup_𝑥∈(0,1)𝜙^′(𝑥≤2𝑄_max)𝑐𝑝−𝑝^′₁⋅sup_𝑥∈(0,1)𝜙^′(𝑥)<𝜀. (10) Therefore, ⟨𝑄⟩_𝜙[𝑝] is stable.
On the other hand, suppose that lim_𝑥→+0𝜙(𝑥)/𝑥∉(0,∞). That is, (i) lim_𝑥→+0𝜙(𝑥)/𝑥=0 or (ii) lim_𝑥→+0𝜙(𝑥)/𝑥=∞. Below, we will examine these cases separately.(i)Consider the following deformation: 𝑝_𝑖=1𝑊−11−𝛿_𝑖1,𝑝^′_𝑖=𝛿1−2𝑝_𝑖+𝛿2𝛿_𝑖1, (11) which are normalized and satisfy ‖𝑝−𝑝^′‖₁=𝛿. We have ^𝑊_𝑖=1𝜙𝑝_𝑖=()𝜙1𝑊−1,𝑊−1_𝑊𝑖=1𝜙𝑝^′_𝑖𝛿=𝜙2+()𝜙1𝑊−1𝛿𝑊−11−2. (12) Difference of the expectation values is calculated as follows: ⟨𝑄⟩_𝜙𝑝−⟨𝑄⟩_𝜙𝑝^′𝑄=−¹𝜙(𝛿/2)𝜙()+()𝜙(+𝛿/2𝑊−1((1/𝑊−1))(1−𝛿/2))_𝑊𝑖=2𝑄_𝑖1−𝜙(𝑊−1((1/𝑊−1))(1−𝛿/2))𝜙()+()𝜙(=𝑊𝛿/2𝑊−1((1/𝑊−1))(1−𝛿/2))𝑊−1𝑄−𝑄₁×𝜙(𝛿/2)/(1−𝛿/2)𝜙(𝛿/2)/(1−𝛿/2)+𝜙((1/(𝑊−1))(1−𝛿/2))/[(1/(𝑊−1))(1−𝛿/2)]_𝑊→∞−−−−−−→𝑄−𝑄₁, (13) since lim_𝑥→+0𝜙(𝑥)/𝑥=0, where 𝑄 is the arithmetic mean, 𝑄=∑^𝑊_𝑖=1𝑄_𝑖/𝑊. Therefore, ⟨𝑄⟩_𝜙[𝑝] is not stable.(ii)Consider the following deformation: 𝑝_𝑖=𝛿_𝑖1,𝑝^′_𝑖=𝛿1−2𝑊𝑝𝑊−1^𝑖+𝛿21,𝑊−1 (14) which are also normalized and satisfy ‖𝑝−𝑝^′‖₁=𝛿. We have ^𝑊_𝑖=1𝜙𝑝_𝑖=𝜙_𝑊𝑖=1(1),𝜙𝑝^′_𝑖𝛿=𝜙1−2+()𝜙𝛿𝑊−121.𝑊−1 (15) Difference of the expectation values is calculated as follows: ⟨𝑄⟩_𝜙𝑝−⟨𝑄⟩_𝜙𝑝^′=𝑄₁𝜙()1−1−𝛿/2𝜙()+()𝜙(−1−𝛿/2𝑊−1((𝛿/2)(1/𝑊−1)))_𝑊𝑖=2𝑄_𝑖𝜙((𝛿/2)(1/(𝑊−1)))𝜙()+()𝜙(=𝑊1−𝛿/2𝑊−1((𝛿/2)(1/𝑊−1)))𝑄𝑊−1¹−𝑄×𝜙((𝛿/2)(1/(𝑊−1)))/[(𝛿/2)(1/(𝑊−1))]𝜙(1−𝛿/2)/(𝛿/2)+𝜙((𝛿/2)(1/(𝑊−1)))/[(𝛿/2)(1/(𝑊−1))]_𝑊→∞−−−−−−→𝑄₁−𝑄, (16) since lim_𝑥→+0𝜙(𝑥)/𝑥=∞. Therefore, ⟨𝑄⟩_𝜙[𝑝]is not stable.

In the above proof, we have employed the specific deformations of the probability distributions as the counterexamples, which are considered in [5]. It is pointed out in [13] that these deformed distributions may experimentally be generated.

Finally, we mention a couple of simple stable examples.

Example 1. 𝜙(𝑥)=𝑒^𝑥−1. (17)

Example 2. 𝜙(𝑥)=ln1+𝑥^𝛼, (18) which yields a stable generalized expectation value, if and only if 𝛼=1.

On the other hand, as mentioned earlier, the 𝑞-expectation value is not stable, since 𝜙(𝑥)=𝑥^𝑞(𝑞>0,𝑞≠1)does not satisfy the condition lim_𝑥→+0𝜙(𝑥)/𝑥∈(0,∞).

In conclusion, we have considered a class of generalized definitions of expectation value that are often employed in nonequilibrium statistical mechanics for complex systems, and have presented the necessary and sufficient condition for such a class to be stable under small deformations of a given arbitrary probability distribution.

Acknowledgment

The work of S. Abe was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.