Thermal and mechanical equilibrium among weakly interacting systems in generalized thermostatistics framework

A.M. Scarfone

REVIEW

Thermal and mechanical equilibrium among weakly interacting systems in generalized thermostatistics framework

A.M. Scarfone

Statistical Mechanics · Submitted 27 Feb 2006 · 24 references

Pith's verdict: A competent but logically gapped derivation of Sharma-Mittal stability conditions: the new two-parameter inequalities are at best sufficient, not necessary, despite the paper's stronger claims.

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arXiv IDcond-mat/0602631 (version 1)

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CitationA.M. Scarfone. "Thermal and mechanical equilibrium among weakly interacting systems in generalized thermostatistics framework." arXiv:cond-mat/0602631 (2006). https://pith.science/paper/MOU7ERU5

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Editor's note

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The paper gives a self-contained variational treatment of equilibrium and stability for two weakly interacting Sharma-Mittal systems, and the explicit two-parameter stability conditions in Eqs. (4.5)-(4.6) are new relative to the cited literature. The temperature and pressure results agree with Abe's earlier definitions, which the author acknowledges; the genuinely new content is the stability analysis and the comparison of entropy vs energy representations. That part is worth attention.

What the paper does well: the physical picture is clear—composability shifts the equilibrium condition from β to β/[1+(1-q)S], and the pressure formula P=-(∂U/∂V)_S is recovered. The variational framework is standard, and the derivation is self-contained rather than assumed. Credit is due for spelling out the two-parameter inequalities explicitly.

The soft spots are real. The worst is the logical step at Eq. (4.3) and again at Eq. (5.5). The second variation of S_{A∪B} is a weighted sum of the two systems' deformed Hessians. The paper moves from 'this sum is negative definite' to 'each individual system must satisfy (4.4)'. That inference is invalid: a positive-weighted sum of two quadratic forms can be negative definite even when one summand is indefinite (take diag(1,-1) plus diag(-3,-1) = diag(-2,-2)). So the per-system conditions are at best sufficient, not necessary, and the paper's stronger language—'the thermodynamical stability conditions' and the conclusion that concavity is neither necessary nor sufficient—outruns the derivation. To make that conclusion hold, the author would need a separate argument, e.g., that stability against an arbitrary partner forces each system's deformed Hessian condition. That is absent.

The printed algebra also has typos: Eq. (3.2) has a sign mismatch in the factorization, Eq. (5.2) uses the same prefactor for both A and B terms, and Eq. (5.5) repeats ∂²U_A in the second volume term instead of ∂²U_B. These are correctable but they make the displayed derivation hard to check.

The reader's worry about 1+(1-q)S > 0 is not actually a gap. From Eq. (2.1), 1+(1-q)S = Z^{1-q} with Z>0, so the sign is automatic. That part is fine.

Who is this for: anyone working with Sharma-Mittal or Tsallis entropies in thermostatistics who needs stability criteria. The corrected version would be a useful reference. As printed, the central claim about non-equivalence with concavity needs reframing as a sufficiency result unless the necessity direction can be proved.

Recommendation: send to peer review, but the referee should push hard on the necessity question and on the equation typos. A revised version stating the conditions as sufficient, with the logical gap openly addressed, is publishable.

Referee report

Significance

If the stability claims were correct, the paper would provide a useful unification: temperature and pressure definitions for the Sharma-Mittal family are derived dynamically rather than postulated, and the relation between nonlinear composability and thermodynamic stability would be clarified for a two-parameter family containing Tsallis and Rényi entropies. The first-order equilibrium derivation is standard and the resulting deformed temperature for the Tsallis case matches earlier definitions. However, the central stability result as printed depends on an invalid inference from definiteness of a weighted sum of Hessians to definiteness of each summand; the paper's claims about non-equivalence with concavity are therefore not established. The paper would need to either prove necessity or explicitly restrict its conclusions to sufficient conditions.

Major objections (3)

[§4, Eqs. (4.3)–(4.6)] The step from Eq. (4.3) to Eqs. (4.4)–(4.6) is not valid as a derivation of necessary stability conditions. Eq. (4.3) states that the second variation of the total entropy is a positive-weighted sum of the two deformed quadratic forms Q_A and Q_B; the text explicitly says the sum is negative definite 'if' each Q is negative definite. The converse inference is used later when Eqs. (4.5)–(4.6) are called 'the thermodynamical stability conditions' and when Section 6 concludes that concavity conditions are not sufficient for stability. A sum of two quadratic forms can be negative definite while one summand is indefinite: for example, diag(1,-1)+diag(-3,-1)=diag(-2,-2). Since the total constraints δU_A+δU_B=0 and δV_A+δV_B=0 leave only the two independent variations appearing in Eq. (4.3), the per-system inequalities (4.4) are at best sufficient. To support the paper's claims, the author must either prove necessity (for example by requiring stability against arbitrary partners) or replace the necessity language by a statement that (4.4) is a sufficient stability condition.
[§5, Eqs. (5.5)–(5.6)] The same logical gap occurs in the energy representation. Eq. (5.5) expresses the second variation of the total energy as a positive-weighted sum of the energy Hessians of A and B, and the text then states that 'from Eq. (5.5) it follows' that each system satisfies the convexity conditions (5.6). Positive definiteness of a weighted sum of two quadratic forms does not imply positive definiteness of each summand; a diagonal example with one indefinite summand and a sufficiently convex partner gives a positive definite sum. Therefore the conclusion that 'TSCs imply the convexity conditions for the energy' is not established. As in §4, the author should either provide an additional argument (e.g., stability against arbitrary partners) or present (5.6) as sufficient.
[§6 and §4 discussion] The paper's central conclusion that 'concavity conditions alone are not necessary nor sufficient conditions for the stability of the equilibrium' is not supported by the preceding derivation. The demonstration that for q<1 concavity implies Eqs. (4.5)–(4.6) and that for q>1 concavity may fail to imply them only concerns the relation between concavity and the sufficient conditions (4.4). It does not establish anything about stability of the composite unless (4.4) is known to be necessary. In particular, the statement that a sub-additive entropy with 'not very pronounced concavity' can violate (4.5)–(4.6) does not show that the equilibrium is unstable. This overreach should be corrected.

Minor objections (6)

[Eq. (3.2)] The prefactor in the second line should be [1+(1-q)S_A][1+(1-q)S_B]; the printed [1+(1-q)S_A][1+(q-1)S_B] = [1+(1-q)S_A][1-(1-q)S_B] is not algebraically equivalent to the preceding line.
[Eq. (5.2)] In the first-order energy variation, the second bracket should contain [1+(1-q)S_B] multiplying ∂U^B/∂S^B, not [1+(1-q)S_A].
[Eq. (5.5)] In the final (δV)^2 term, the second summand should be ∂²U^B/∂(V^B)^2, not a second copy of ∂²U^A/∂(V^A)^2.
[After Eq. (3.2)] The assertion 1+(1-q)S>0 is used without proof; it follows immediately from Eq. (2.1) and Eq. (2.8) because 1+(1-q)S = Z^{1-q}>0, so this line should be added.
[Eqs. (2.17)–(2.18)] The wording 'for a stable equilibrium configuration, implies' is backwards: Eq. (2.18) is a consequence of Eq. (4.5), not an independent input, and this dependence should be stated explicitly.
[Eq. (4.7)] The quantity B_{m} is introduced as 'a negative definite quantity for a concave entropy', but its role in Eq. (4.6) and the sign of the right-hand side are not explained; a short derivation would help the reader.

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The paper

The paper claims

This paper shows that equilibrium between weakly interacting systems sharing the Sharma-Mittal entropy is controlled by a deformed inverse temperature, and that thermodynamic stability requires deformed concavity conditions, not ordinary…

The reading

The paper works out how two weakly interacting systems reach equilibrium when both are described by the same two-parameter Sharma-Mittal entropy, a family that includes Rényi, Tsallis, and Boltzmann-Gibbs entropies as special cases. It identifies the quantity that actually controls heat exchange as a deformed inverse temperature, $\tilde{\beta} = \beta/(1+(1-q)S)$, and shows that equilibrium is reached when $\tilde{\beta}$ is the same in both systems, with temperature $T = 1/\tilde{\beta}$. Pressure is defined analogously and is shown to still equal $-\partial U/\partial V$. The central result is that thermodynamic stability is not equivalent to ordinary concavity of the entropy: for super-additive ($q<1$) systems concavity is sufficient but not necessary, while for sub-additive ($q>1$) systems concavity is not sufficient. A reader should care because this gives a unifying derivation of temperature, pressure, and stability for a large class of non-extensive entropies without assuming a particular one-parameter form.

What carries the argument

The deformed inverse temperature $\tilde{\beta} = \beta/(1+(1-q)S)$, where $S$ is the Sharma-Mittal entropy and $\beta = (\partial S/\partial U)_V$, is the scalar that controls heat exchange and defines temperature via $T = 1/\tilde{\beta}$. The companion deformed Hessian $S_{XY} = \partial^2 S/\partial X\partial Y - \frac{1-q}{1+(1-q)S} \frac{\partial S}{\partial X} \frac{\partial S}{\partial Y}$ carries the stability analysis: the conditions $S_{UU}<0$ and $S_{UU}S_{VV}-(S_{UV})^2>0$ are the paper's thermodynamic stability conditions. Both objects arise from the non-linear composability rule $S_{A\cup B} = S_A + S_B + (1-q) S_A S_B$, which injects the $q$-dependence into every equilibrium and stability statement.

Extended reading notes

Core claim

For two statistically independent systems A and B described by the same Sharma-Mittal entropy and exchanging only weakly, the first-order variation of the composite entropy decomposes so that the evolution is controlled by the two scalars $\tilde{\beta} = \beta/(1+(1-q)S)$ and $P = T/(1+(1-q)S)(\partial S/\partial V)_U$. Equilibrium is stated by $\tilde{\beta}_A = \tilde{\beta}_B$ and $P_A = P_B$, with heat flowing from lower to higher $\tilde{\beta}$ and volume flowing from higher to lower pressure. Expanding the entropy to second order gives the stability conditions $S_{UU}<0$ and $S_{UU}S_{VV}-(S_{UV})^2>0$, where $S_{XY}$ is the second derivative of entropy corrected by a term proportional to $(1-q)/(1+(1-q)S)$ times first-derivative products. These reduce to ordinary concavity only in the $q=1$ limit. In the energy representation, assuming linear composition of energy, the same equilibrium conditions appear, and stability is exactly the convexity of the energy. The paper concludes that the form of the composability rule of the relevant thermodynamic quantity determines whether stability can be stated as plain concavity or convexity.

Load-bearing premise

The paper assumes throughout that $1+(1-q)S$ is positive; all directions of heat flow, the volume equilibrium condition, and the stability weights rest on that inequality, which is stated but not proven.

Editorial extensions

If this is right

Heat always flows from the system with smaller $\tilde{\beta}$ to the system with larger $\tilde{\beta}$, and equilibrium is reached only when $\tilde{\beta}$ is equal in both systems.
The pressure defined through the deformed entropy is identical to the standard $-\partial U/\partial V$, so mechanical equilibrium retains its familiar form even for non-extensive entropies.
Thermodynamic stability requires the deformed Hessian conditions $S_{UU}<0$ and $S_{UU}S_{VV}-(S_{UV})^2>0$, which are not equivalent to ordinary concavity when $q\neq 1$.
For super-additive systems ($q<1$), ordinary concavity of the entropy is sufficient for stability and a small convexity can still be stable, while for sub-additive systems ($q>1$) concavity is insufficient.
In the energy representation, with a linear composition rule for energy, stability is equivalent to convexity of the energy, independent of the entropy's composability rule.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the denominator $1+(1-q)S$ can change sign within the accessible energy and volume range, the deformed temperature and pressure would diverge or flip sign, so locating such crossings in concrete models would test the domain of validity as a separate question from entropy concavity.
The super-stability found for $q<1$ suggests that certain entropy functions with mildly convex regions could still describe thermodynamically stable systems, potentially relevant for small or finite-size systems where entropy is not globally concave.
The same deformed-temperature formalism can be applied to non-extensive kinetic models; one could test whether heat flow in a two-temperature Sharma-Mittal gas follows the sign of the $\tilde{\beta}$ difference even when the ordinary $\beta$ predicts the opposite direction.

Original abstract

We consider two statistically independent systems described by the same entropy belonging to the two-parameter family of Sharma-Mittal. Assuming a weak interaction among the systems, allowing in this way an exchange of heat and work, we analyze, both in the entropy representation and in the energy representation, the evolution toward the equilibrium. The thermodynamics evolution is controlled by two scalar quantities identified with the temperature and the pressure of the system. The thermodynamical stability conditions of the equilibrium state are analyzed in both representations. Their relationship with the concavity conditions for the entropy and with the convexity conditions for the energy are spotlighted.

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We consider two statistically independent systems described by the same entropy belonging to the two-parameter family of Sharma-Mittal. Assuming a weak interaction among the systems, allowing in this way an exchange of heat and work, we analyze, both in the entropy representation and in the energy representation, the evolution toward the equilibrium. The thermodynamics evolution is controlled by two scalar quantities identified with the temperature and the pressure of the system. The thermodynamical stability conditions of the equilibrium state are analyzed in both representations. Their relationship with the concavity conditions for the entropy and with the convexity conditions for the energy are spotlighted.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider two statistically independent systems described by the same entropy belonging to the two-parameter family of Sharma-Mittal.	contribution 0.55	0.50	0.03	asserted 0.92
s1	Assuming a weak interaction among the systems, allowing in this way an exchange of heat and work, we analyze, both in the entropy representation and in the energy representation, the evolution toward the equilibrium.	method 0.71	0.86	0.06	asserted 0.93
s2	The thermodynamics evolution is controlled by two scalar quantities identified with the temperature and the pressure of the system.	result 0.96	0.81	0.26	asserted 1.00
s3	The thermodynamical stability conditions of the equilibrium state are analyzed in both representations.	contribution 0.89	0.89	0.08	asserted 0.95
s4	Their relationship with the concavity conditions for the entropy and with the convexity conditions for the energy are spotlighted.	interpretation 0.53	0.83	0.14	asserted 0.98

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References

Works cited (24)

[1]

H.B. Callen, Thermodynamics and an Introduction to Thermostatistics , (Wiley, New York, 1985)

View on Pith 1985
[2]

G.H. Wannier, Statistical Physics, (Dover pubblications, Inc., New York, 1966)

View on Pith 1966
[3]

B.D. Sharma and D.P. Mittal, J. Math. Sci. 10, 28 (1975)

View on Pith 1975
[4]

A. R´ enyi, Probability theory , (North-Holland Publ. Company, Amsterdam, 1970)

View on Pith 1970
[5]

C. Tsallis, J. Stat. Phys. 52, 479 (1988)

View on Pith 1988
[6]

P.T. Landberg and V. Vedral, Phys. Lett. A 247, 211 (1998)

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[7]

T.D. Frank and A.R. Plastino, Eur. Phys. J. B 30, 543 (2002)

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[8]

T.D. Frank and A. Daﬀertshofer, Physica A 285, 351 (2000)

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[9]

K.S. Fa and E.K. Lenzi, Chaos, Solitons and Fractals 20, 227 (2004)

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[10]

E.K. Lenzi, M.K. Lenzi, H. Belich, and L.S. Lucena, Phys . Lett. A 292, 315 (2002)

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[11]

M. Masi, Phys. Lett. A 338, 217 (2005). 14

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[12]

M. Czachor and J. Naudts, Phys. Lett. A 298, 369 (2002)

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[13]

S. Abe, Physica A 300, 417 (2001)

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[14]

S. Abe, S. Martinez, F. Pennini, and A. Plastino, Phys. L ett. A 278, 249 (2001); Phys. Lett. A 281, 126 (2001)

View on Pith 2001
[15]

T. Wada, Phys. Lett. A 297, 334 (2002); Continuum Mech. Thermodyn. 16, 263 (2004); Physica A 340, 126 (2004)

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[16]

A.M. Scarfone and T. Wada, Phys. Rev. E 71, 051103 (2005)

View on Pith 2005
[17]

J.D. Ramshaw, Phys. Lett. A 198, 119 (1995)

View on Pith 1995
[18]

C. Tsallis and E. Brigatti, Continuum Mech. Thermodyn. 16, 223 (2004)

View on Pith 2004
[19]

C. Tsallis, R.S. Mendes, and A.R. Plastino, Physica A 261, 534 (1998)

View on Pith 1998
[20]

A.M. Scarfone and T. Wada “Canonical partition function for anomalous systems described by the κ-entropy”, arXiv:cond-mat/0509364; Prog. Theor. Phys. Supl. (2006), in press

View on Pith Pith review arXiv 2006
[21]

A.M. Scarfone, “Legendre structure of the thermostatistics theory based o n the Sharma-Taneja-Mittal entropy”, arXiv:cond-mat/0509774; Physica A (2006), in press

View on Pith Pith review arXiv 2006
[22]

R. Toral, Physica A 317, 209 (2003)

View on Pith 2003
[23]

G. Kaniadakis, M. Lissia, and A.M. Scarfone, Physica A 340, 41 (2004); Phys. Rev. E 71, 046128 (2005)

View on Pith 2004
[24]

Q.A. Wang, L. Nivanen, A. Le M´ ehaut´ e, and M. Pezeril, J . Phys. A: Math. Gen. 35, 7003 (2002); Eur. Phys. Lett. 65, 606 (2004). 15

View on Pith 2002

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

H.B. Callen, Thermodynamics and an Introduction to Thermostatistics , (Wiley, New York, 1985)

View on Pith 1985

[2] [2]

G.H. Wannier, Statistical Physics, (Dover pubblications, Inc., New York, 1966)

View on Pith 1966

[3] [3]

B.D. Sharma and D.P. Mittal, J. Math. Sci. 10, 28 (1975)

View on Pith 1975

[4] [4]

A. R´ enyi, Probability theory , (North-Holland Publ. Company, Amsterdam, 1970)

View on Pith 1970

[5] [5]

C. Tsallis, J. Stat. Phys. 52, 479 (1988)

View on Pith 1988

[6] [6]

P.T. Landberg and V. Vedral, Phys. Lett. A 247, 211 (1998)

View on Pith 1998

[7] [7]

T.D. Frank and A.R. Plastino, Eur. Phys. J. B 30, 543 (2002)

View on Pith 2002

[8] [8]

T.D. Frank and A. Daﬀertshofer, Physica A 285, 351 (2000)

View on Pith 2000

[9] [9]

K.S. Fa and E.K. Lenzi, Chaos, Solitons and Fractals 20, 227 (2004)

View on Pith 2004

[10] [10]

E.K. Lenzi, M.K. Lenzi, H. Belich, and L.S. Lucena, Phys . Lett. A 292, 315 (2002)

View on Pith 2002

[11] [11]

M. Masi, Phys. Lett. A 338, 217 (2005). 14

View on Pith 2005

[12] [12]

M. Czachor and J. Naudts, Phys. Lett. A 298, 369 (2002)

View on Pith 2002

[13] [13]

S. Abe, Physica A 300, 417 (2001)

View on Pith 2001

[14] [14]

S. Abe, S. Martinez, F. Pennini, and A. Plastino, Phys. L ett. A 278, 249 (2001); Phys. Lett. A 281, 126 (2001)

View on Pith 2001

[15] [15]

T. Wada, Phys. Lett. A 297, 334 (2002); Continuum Mech. Thermodyn. 16, 263 (2004); Physica A 340, 126 (2004)

View on Pith 2002

[16] [16]

A.M. Scarfone and T. Wada, Phys. Rev. E 71, 051103 (2005)

View on Pith 2005

[17] [17]

J.D. Ramshaw, Phys. Lett. A 198, 119 (1995)

View on Pith 1995

[18] [18]

C. Tsallis and E. Brigatti, Continuum Mech. Thermodyn. 16, 223 (2004)

View on Pith 2004

[19] [19]

C. Tsallis, R.S. Mendes, and A.R. Plastino, Physica A 261, 534 (1998)

View on Pith 1998

[20] [20]

A.M. Scarfone and T. Wada “Canonical partition function for anomalous systems described by the κ-entropy”, arXiv:cond-mat/0509364; Prog. Theor. Phys. Supl. (2006), in press

View on Pith Pith review arXiv 2006

[21] [21]

A.M. Scarfone, “Legendre structure of the thermostatistics theory based o n the Sharma-Taneja-Mittal entropy”, arXiv:cond-mat/0509774; Physica A (2006), in press

View on Pith Pith review arXiv 2006

[22] [22]

R. Toral, Physica A 317, 209 (2003)

View on Pith 2003

[23] [23]

G. Kaniadakis, M. Lissia, and A.M. Scarfone, Physica A 340, 41 (2004); Phys. Rev. E 71, 046128 (2005)

View on Pith 2004

[24] [24]

Q.A. Wang, L. Nivanen, A. Le M´ ehaut´ e, and M. Pezeril, J . Phys. A: Math. Gen. 35, 7003 (2002); Eur. Phys. Lett. 65, 606 (2004). 15

View on Pith 2002