The Uniqueness of the Integration Factor Associated with the Exchanged Heat in Thermodynamics

Fundamental Research 1 (2021) 6–9

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The uniqueness of the integration factor associated with the exchanged heat in thermodynamics

Yu-Han Ma a,b, Hui Dong b, Hai-Tao Quan c, Chang-Pu Sun a,b,∗ a Beijing Computational Science Research Center , Beijing 100193 , China b Graduate School of China Academy of Engineering Physics , No. 10 Xibeiwang East Road, Haidian District , Beijing 100193 , China c School of Physics , Peking University , Beijing 100871 , China

a r t i c l e i n f o a b s t r a c t

Keywords: State functions play important roles in thermodynamics. Diﬀerent from the process function, such as the ex- Integration factor changed heat 𝛿𝑄 and the applied work 𝛿𝑊 , the change of the state function can be expressed as an exact diﬀer-

Process function ential. We prove here that, for a generic thermodynamic system, only the inverse of the temperature, namely 1∕ 𝑇 ,

Exchanged heat can serve as the integration factor for the exchanged heat 𝛿𝑄 . The uniqueness of the integration factor invalidates Thermodynamic entropy any attempt to deﬁne other state functions associated with the exchanged heat, and in turn, reveals the incorrect- Uniqueness theorem 2 ness of deﬁning the entransy 𝐸 𝑣ℎ = 𝐶 𝑉 𝑇 ∕2 as a state function by treating T as an integration factor. We further show the errors in the derivation of entransy by treating the heat capacity 𝐶 𝑉 as a temperature-independent constant.

1. Introduction prove that only one factor, namely 1∕ 𝑇 , can serve as the integration factor to convert the exchanged heat into the state function. Such unique- State functions, e.g., the internal energy and the entropy, in ther- ness invalidates any attempt to find new state functions associated with modynamics characterize important features of the system in a thermal the exchanged heat 𝛿𝑄 . equilibrium state [1] . Physically, some quantities are process-dependent It is worth mentioning that the introduction of the so-called “state and thus cannot be treated as state functions, e.g., the exchanged heat function ”, the entransy defined via 𝛿𝐸 𝑣ℎ = 𝑇 𝛿𝑄 [5] , in the realm of the 𝛿𝑄 and the applied work 𝛿𝑊 . An integration factor 𝑓 can be utilized heat transfer is an example of such attempt. The entransy is claimed to be to convert the process function, e.g., the exchanged heat, into an exact a “state function ”of a system characterizing its potential of heat transfer differential (the change of a state function). Mathematically, the require- [5] . Since its appearance, it has triggered a lot of debates [6-10] over ment of the state function can be expressed as follows: the change of the its validity as state function [6] as well as its usefulness [7-10] in the state function remains unchanged with any topological variation of the practical application in the field of heat transfer. In this paper, from the integration path on the parameter space [2] . In addition, the number of fundamental principles of thermodynamics [11-13] and with the help the integration factors is usually limited for any system with more than of statistical mechanics, we reveal the improperness and incorrectness two thermodynamic variables. Such number of the integration factor is of such a concept. further reduced in order to define a universal state function without the The rest of the paper is organized as follow. In Sec. 2, we prove the dependence on the system characteristics [2] . It is a common sense that uniqueness of the integration factor 1∕ 𝑇 and the resultant entropy as the inverse of the temperature, i.e., 1∕ 𝑇 , serves as the integration factor the state function, then we further discuss the errors in the definition of for the exchanged heat 𝛿𝑄 , and thus a state function, the entropy can the entransy. The conclusions are given in Sec. 3. be defined. A relevant question arises here: is there any other universal integration factor associated with the exchanged heat 𝛿𝑄 for an arbitrary system? Several attempts on this issue have been made for both 2. Proof of the uniqueness of the integration factor 𝟏∕ 𝑻 specific systems [3] and generic systems [4] . However, the uniqueness of the integration factor of interest for a generic thermodynamic system In this section, we will prove the uniqueness of the integration fac- remains unexplored. In our current paper, from the first principle, we tor associated with the exchanged heat 𝛿𝑄 . As an illustration, we first demonstrate the result in the ideal gas with the internal energy 𝑈, the temperature 𝑇 , the volume 𝑉 , and the pressure 𝑃 .

∗ Corresponding author. E-mail address: [email protected] (C.-P. Sun).

https://doi.org/10.1016/j.fmre.2020.11.003

Available online 10 December 2020 2667-3258/© 2021 The Authors. Publishing Services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) Y.-H. Ma, H. Dong, H.-T. Quan et al. Fundamental Research 1 (2021) 6–9

Having shown the uniqueness of the integration factor associated with the exchanged heat for the classical ideal gas, in the following, we will prove a theorem on the uniqueness of the integration factor associated with 𝛿𝑄 for a generic thermodynamic system.

Theorem. For a generic thermodynamic system, the universal thermodynamic state function Λ can be deﬁned as 𝑑Λ= 𝑓( 𝑇 , 𝜆) 𝛿𝑄 , if and only if 𝑓( 𝑇 , 𝜆) = α∕ 𝑇 with 𝛼, a system-independent constant.

Proof. For a generic thermodynamic system with the internal energy 𝑈, the first law of thermodynamic law reads 𝛿𝑄 = 𝑑𝑈 − 𝛿𝑊 = 𝑑 𝑈 − 𝑌 𝑑 𝜆, (9) where 𝑌 and 𝜆 are the generalized force and the generalized displace- ment, respectively. Similar to the discussions for the ideal gas system, the condition for 𝛿Λ= 𝑓( 𝑇 , 𝜆) 𝛿𝑄 to be an exact differential ( Λ to be a Fig. 1. The evolution path of the ideal gas in the parameter space 𝑇 − 𝑉 . state function) is: For an arbitrary loop in the 𝑇 − 𝑉 space, we always have According to the first law of thermodynamics, the changed heat of ΔΛ = 𝑓 ( 𝑇 , 𝜆) 𝛿𝑄 = 𝑓 ( 𝑇 , 𝜆)( 𝑑 𝑈 − 𝑌 𝑑 𝜆) = 0 . (10) the gas reads 𝛿𝑄 = 𝑑𝑈 − 𝛿𝑊 , which can be further written as (with the ∮ ∮ work done on the ideal gas 𝛿𝑊 = − 𝑃 𝑑𝑉 ) Noticing the relation for the internal energy 𝛿𝑄 𝑑 𝑈 𝑃 𝑑 𝑉 . = + (1) 𝜕𝑈 𝜕𝑈 𝑑 𝑈 = 𝑑 𝑇 + 𝑑 𝜆, (11) 𝜕𝑇 𝜕𝜆 An infinitesimal change of a generic thermodynamic function Λ can be defined as 𝛿Λ= 𝑓( 𝑇 ) 𝛿𝑄 , namely, Eq. (10) can be rewritten in the form of the surface integration with Green’s theorem as 𝛿Λ= 𝑓 ( 𝑇 )( 𝑑 𝑈 + 𝑃 𝑑 𝑉 ) . (2) { [ ] [ ( )]} 𝜕 𝜕𝑈 𝜕 𝜕𝑈 𝑓 𝑇 , 𝜆 𝑓 𝑇 , 𝜆 𝑌 𝑑 𝜆𝑑 𝑇 . 𝑇 , 𝑉 ΔΛ = ( ) − ( ) − = 0 (12) As illustrated in Fig. 1, the state ( 0 0 ) at point A can be connected ∫∫ 𝜕𝜆 𝜕𝑇 𝜕𝑇 𝜕𝜆 𝑙 𝑎𝑛𝑑 𝑙 𝑇 , 𝑉 𝑇 𝑉 by two paths ( 1 2 ) to the state ( 𝑓 𝑓 ) at point B in the − 𝑙 𝑙 The above equation is valid irrespective of the integration intervals. ∫ 1 𝛿 ∫ 2 𝛿 space. If Λ is a state function, it is required that A →B Λ= A →B Λ [2]. Hence, we have Accordingly, the loop integral of 𝛿Λ in the 𝑇 − 𝑉 space is strictly zero, [ ] [ ( )] 𝜕 𝜕𝑈 𝜕 𝜕𝑈 namely, 𝑓 ( 𝑇 , 𝜆) − 𝑓 ( 𝑇 , 𝜆) − 𝑌 = 0 , (13) 𝜕𝜆 𝜕𝑇 𝜕𝑇 𝜕𝜆 𝑙 𝑙 𝑙 𝑙 1 2 1 2 𝛿Λ= ∫ 𝛿Λ+ ∫ 𝛿Λ= ∫ 𝛿Λ− ∫ 𝛿Λ= 0 . (3) which can be further simplified as ∮ → → → → ( ) A B B A A B A B 𝜕𝑈 𝜕𝑓 ( 𝑇 , 𝜆) 𝜕𝑌 𝜕𝑈 𝜕𝑓 ( 𝑇 , 𝜆) + 𝑓 ( 𝑇 , 𝜆) + 𝑌 − = 0 . (14) With Eq. (2) , the above equation can be specifically written as 𝜕𝑇 𝜕𝜆 𝜕𝑇 𝜕𝜆 𝜕𝑇 ( ) 𝑛𝑅𝑇 In statistical mechanics, the internal energy and the generalized force ΔΛ = 𝛿Λ= 𝑓 ( 𝑇 ) 𝐶 𝑉 𝑑 𝑇 + 𝑑 𝑉 = 0 , (4) ∮ ∮ 𝑉 can be written as [1] 𝑗 𝑁 where we have used 𝑑 𝑈 = 𝐶 𝑉 𝑑 𝑇 and the equation of state of the ideal ∑= 𝑈 𝑝 𝐸 , gas 𝑃 𝑉 = 𝑛𝑅𝑇 . Here, 𝐶 𝑉 is the heat capacity at a constant volume, 𝑛 is = 𝑗 𝑗 (15) 𝑗 the number of moles of the gas, and 𝑅 is the gas constant. With Green’s =1 theorem, the loop integral in Eq. (3) can be rewritten in the form of the and ( ) surface integration as 𝑗= 𝑁 ∑ 𝑑 𝐸 𝑗 { [ ]} 𝑌 𝑝 , 𝜕 [ ] 𝜕 𝑛𝑅𝑇 = 𝑗 𝑑𝜆 (16) ΔΛ = 𝑓 ( 𝑇 )𝐶 𝑉 − 𝑓 ( 𝑇 ) 𝑑 𝑉 𝑑 𝑇 = 0 . (5) 𝑗=1 ∫∫ 𝜕𝑉 𝜕𝑇 𝑉 𝐸 𝐸 𝜆 𝑗 , 𝑁 𝑗 1 This equation is valid irrespective of the integral intervals in the 𝑇 − where 𝑗 = 𝑗 ( ) ( = 1 2 … ) is the th energy level of the system , 𝜆 𝑉 space. Hence, we have and it is -dependent. 𝛽𝐸 𝜆 𝛽𝐸 𝜕 𝐶 [ ] 𝑒 − 𝑗 ( ) 𝑒 − 𝑗 𝑉 𝜕 𝑛𝑅𝑇 𝑝 ≡ 𝑓 ( 𝑇 ) − 𝑓 ( 𝑇 ) = 0 . (6) 𝑗 = ∑𝑗 𝑁 𝛽𝐸 𝜆 (17) 𝜕𝑉 𝜕𝑇 𝑉 = 𝑒 − 𝑗 ( ) 𝑍 𝑗=1 The fact 𝜕 𝐶 ∕ 𝜕𝑉 = 0 for the ideal gas further simplifies the above 𝑉 is the corresponding thermal equilibrium distribution of the system on equations to 𝑗 𝛽 𝑘 𝑇 [ ] the -th energy level with = 1∕( 𝐵 ) as the inverse temperature and 𝜕 𝑛𝑅𝑇 𝑘 𝑓 ( 𝑇 ) = 0 , (7) 𝐵 as the Boltzmann constant. Combining Eqs. (15)–(17), we find 𝜕𝑇 𝑉 ( ) 𝑗= 𝑁 2 2 ∑ − 𝛽𝐸 𝑗 ⟨𝐸 ⟩ − 𝑈 𝜕𝑈 𝜕 𝑒 𝑗 whose solution can be uniquely determined as = 𝐸 𝑗 = ≡ 𝐶 𝜆, (18) 𝜕𝑇 𝜕𝑇 𝑍 2 𝛼 𝑗 𝑘 𝐵 𝑇 𝑓 ( 𝑇 ) = , (8) =1 𝑇 ( ) ( ) 𝑗 𝑁 𝑗 𝑁 ∑= − 𝛽𝐸 𝑗 𝑑 𝐸 ∑= 𝑑 𝐸 𝛼 𝑇 𝜕𝑌 𝜕 𝑒 𝑗 𝛽 𝑗 with being an arbitrary constant independent of the temperature . = = 𝑝 𝑗 𝐸 𝑗 − 𝑈𝑌 , (19) 𝜕𝑇 𝜕𝑇 𝑍 𝑑𝜆 𝑇 𝑑𝜆 Without losing generality, we choose 𝛼 = 1 , i.e., 𝛿Λ= 𝛿𝑄 ∕ 𝑇 . Thus, 𝛿Λ is 𝑗=1 𝑗=1 nothing but the change of the thermodynamic entropy 𝑑𝑆. In summary, we prove that for the classical ideal gas only one state function associ- 1 For simplicity, we consider a system with discrete energy levels. But it is ated with the exchanged heat, namely the entropy, can be defined. A straightforward to extend our discussions to systems with continuous energy similar proof for such an ideal gas system was proposed by Weiss [3] . spectrum.

7 Y.-H. Ma, H. Dong, H.-T. Quan et al. Fundamental Research 1 (2021) 6–9

2 and 𝐶 𝑉 𝑇 ∕2 , with the analogy to the definition of the energy of the elec- ( ) 𝑗= 𝑁 𝛽𝐸 𝑗= 𝑁 tronic capacity. To assure the equivalence of the two definitions above, 𝜕𝑈 𝜕 ∑ 𝑒 − 𝑗 ∑ 𝑑 𝐸 𝑗 = 𝐸 𝑗 = 𝑌 + 𝛽𝑈𝑌 − 𝛽 𝑝 𝑗 𝐸 𝑗 . (20) the capacity at a constant volume is assumed as a constant to simplify 𝜕𝜆 𝜕𝜆 𝑍 𝑑𝜆 𝑇 𝑗=1 𝑗=1 𝐸 ∫ 𝐶 𝑇 𝑇 𝑑𝑇 𝐶 𝑇 2 . the integration 𝑣ℎ = 0 𝑉 ( ) = 𝑉 ∕2 However, such assump- Substituting the above three relations into Eq. (14) , we obtain tion is only valid for some systems, such as classical ideal gas, in the ( ) high temperature regime, but not valid for any real solid-state materi- 𝜕𝑓 ( 𝑇 , 𝜆) 𝜕𝑓 ( 𝑇 , 𝜆) 𝑓 ( 𝑇 , 𝜆) 𝐶 𝜆 + Θ + = 0 , (21) als in an arbitrary circumstance. One typical heat capacity 𝐶 𝑉 as the 𝜕𝜆 𝜕𝑇 𝑇 function of temperature 𝑇 is known as the Debye’s law, which shows 3 where that, in the low temperature regime of 𝑇 ≪ Θ𝐷 , 𝐶 𝑉 ∝ 𝑇 rather than a ( ) 𝑗= 𝑁 ∑ 𝑑 𝐸 𝑗 𝜕𝑌 constant [14]. Here Θ𝐷 is the Debye temperature. Such oversimplified Θ ≡ 𝛽 𝑝 𝑗 𝐸 𝑗 − 𝑈𝑌 = 𝑇 (22) 𝑑𝜆 𝜕𝑇 assumption prevents the practical application. 𝑗=1 is determined by the equation of state 𝑌 = 𝑌 ( 𝑇 , 𝜆) of the system. 3. Conclusions With the assumption of the factorized structure 𝑓( 𝑇 , 𝜆) = 𝑔( 𝑇 ) ℎ ( 𝜆) , Eq. (21) becomes ( ) In summary, it is a common sense that 1∕ 𝑇 is an integration factor as- 𝑑 lnℎ ( 𝜆) 𝑑 ln 𝑔 ( 𝑇 ) 1 𝛿𝑄 𝐶 𝜆 + Θ + = 0 . (23) sociated with the exchanged heat . The uniqueness of this integration 𝑑𝜆 𝑑𝑇 𝑇 factor has been proven for some specific systems with given equations The solution to Eq. (23) follows: of state [3] . In this paper, from the perspective of statistical mechanics, ( ) we prove the uniqueness of 1∕ 𝑇 as the integration factor associated with 𝑑 ln𝑔 ( 𝑇 ) 1 𝐶 𝜆 𝑑 lnℎ ( 𝜆) + = 𝜇 = − (24) the exchanged heat 𝛿𝑄 for a generic thermodynamic system without re- 𝑑𝑇 𝑇 Θ 𝑑𝜆 ferring to its equation of state. Such a theorem prevents the possibility where 𝜇 = 𝜇( 𝑇 ) is a function of 𝑇 independent of 𝜆. The solution to of defining any new state function associated with the exchanged heat Eq. (24) is other than the entropy. With this theorem, we clearly exclude the possi- 𝜇 ∫ Θ 𝑑𝜆 𝑔 bility of the entransy as a state function in thermodynamics. In addition, − 𝐶 0 ∫ 𝜇𝑑𝑇 ℎ ( 𝜆) = ℎ 𝑒 𝜆 , 𝑔 ( 𝑇 ) = 𝑒 . (25) 0 𝑇 we have also shown errors in the derivation of the entransy with the false

And the integration factor 𝑓( 𝑇 , 𝜆) can be explicitly written as assumption of a temperature-independent heat capacity. We conclude that the entransy cannot be a state function in thermodynamics and the 𝛼 − 𝜇 ∫ Θ 𝑑 𝜆+ ∫ 𝜇𝑑 𝑇 𝑓 ( 𝑇 , 𝜆) = 𝑒 𝐶 𝜆 . (26) false assumption in the derivation prevents its practical applications in 𝑇 heat transfer in any real materials. 𝑔 , ℎ 𝑇 𝜆 𝛼 Here, 0 0 are integral constants independent of and , and = 𝑔 ℎ 0 0 . In order to be consistent with the factorized structure assumption for 𝑓 ( 𝑇 , 𝜆), Θ∕ 𝐶 𝜆 also needs to have a factorized structure of 𝑇 and 𝜆. In Declaration of Competing Interest the equation above, the form of 𝑓( 𝑇 , 𝜆) is not unique due to the multiple choice of the function 𝜇( 𝑇 ) . The dependence of 𝑓( 𝑇 , 𝜆) on Θ∕ 𝐶 𝜆 with the The authors declared that they have no conflicts of interest to this specific thermodynamic heat capacity and the equation of state prohibit work. the definition of the universal quantity. Therefore, 𝜇 can only be set to be 𝜇 = 0 to make Λ a universal state function

𝛼 Acknowledgments 𝑓 ( 𝑇 , 𝜆) = . (27) 𝑇 Y. H. Ma is grateful to R. X. Zhai and Z. Y. Fei for helpful dis-

We have proven the uniqueness theorem of the integration factor cussions. This work was supported by the National Natural Science associated with the exchanged heat. Without loss of generality we set Foundation of China (NSFC) (Grants No. 11534002 , No. 12088101 , 𝛼 = 1, and we obtain No. U1530402 , No. U1930403 , No. 11775001 , No. 11534002 , and No. 𝛿𝑄 𝛿Λ= = 𝑑𝑆, (28) 11825001), and the National Basic Research Program of China (Grants 𝑇 No. 2016YFA0301201 ). which indicates that, associated with the exchanged heat 𝛿𝑄 , the entropy deﬁned via 𝑑𝑆 = 𝛿𝑄 ∕ 𝑇 is the only universal state function. The References existence of the integration factor 1∕ 𝑇 for the exchanged heat is known in thermodynamics. We have shown no other integration factors exist [1] K. Huang , Statistical Mechanics, 2nd ed., John Wiley & Sons, New York, 1987 . for the exchanged heat 𝛿𝑄 . [2] V.V. Sychev , in: The Diﬀerential Equations of Thermodynamics, 2nd ed., MIR Pub-

The entransy 𝛿𝐸 𝑣ℎ = 𝑇 𝛿𝑄 is introduced as a “state function ”for the lishers Moscow, 1991, pp. 19–26. [3] V.C. Weiss , The uniqueness of clausius’ integrating factor, Am. J. Phys. 8 (2006) 74 . purpose of optimizing the heat transfer [5]. The supporters of the en- [4] H. Margenau , G.M. Murphy , in: The Mathematics of Physics and Chemistry, D. Van transy claim it as a new state function by regarding a new integration Nostradn Company, 1956, pp. 26–31 . factor T , which contradicts the theorem. The theorem directly excludes [5] Z.Y. Guo , H.Y. Zhu , X.G. Liang , Entransy —a physical quantity describing heat trans-

fer ability, Int. J. Heat Mass Transf. 50 (2007) 2345–2556. the entransy as a state function. Clearly, the entransy is essentially dif- [6] A. Bejan , Letter to the editor on “temperature-heat diagram analysis method for ferent from the well-deﬁned thermodynamic state quantities such as the heat recovery physical adsorption refrigeration cycle —taking multi stage cycle as internal energy, the free energy, and the entropy. Thus, we conclude that an example ”, Int. J. Refrig. 90 (2018) 277 . the entransy introduced in Ref. [5] cannot be regarded as a fundamental [7] M.M. Awad, Discussion: “entransy is now clear, J. Heat Transf. 136 (2014) 095502. [8] D.P. Sekulic , E. Sciubba , M.J. Moran , Entransy: a misleading concept for the analysis thermodynamic quantity. And the entransy is introduced for the system and optimization of thermal systems, Energy 80 (2015) 251 . with a ﬁxed volume [5] . Such assumption leaves the temperature 𝑇 as [9] H. Herwig , Do we really need ’entransy’? A critical assessment of a new quantity in the only variable. It is meaningless to talk about the state function of a heat transfer analysis, J. Heat Transf. 136 (2014) 045501. [10] S. Oliveria , L. Milanez , Equivalence between the application of entransy and entropy single-valued function since in thermodynamics a state function is ex- generation, Int. J. Heat Mass Transf. 79 (2014) 518 . clusively a function of two or more variables. [11] C. Carathéodory , Untersuchungen über die grundlagen der thermodynamik, Math.

For the case of the single-valued function, the entransy is written ex- Ann. 67 (1909) 355. [12] H.A. Buchdahl , On the principle of carathéodory, Am. J. Phys. 17 (1949) 41 . 𝐸 ∫ 𝑇 𝐶 𝑇 𝑇 𝑑𝑇 plicitly by the integration 𝑣ℎ = 0 𝑉 ( ) . Alternatively, the deﬁni- [13] H.A. Buchdahl , On the theorem of carathéodory, Am. J. Phys. 17 (1949) 44 . tion of the entransy [5] via the internal energy is given as 𝐸 𝑣ℎ = 𝑈𝑇 ∕2 = [14] C. Kittel , Introduction to Solid State Physics vol. 8, Wiley, New York, 1976 .

8 Y.-H. Ma, H. Dong, H.-T. Quan et al. Fundamental Research 1 (2021) 6–9

Dr. Yu-Han Ma, graduated from Beijing Normal University Professor Chang-Pu Sun, obtained his Ph.D. in 1992 at the with a bachelor’s degree in 2015, and obtained his Ph.D. at Chern Institute of Mathematics in Nankai University, currently Beijing Computing Science Research Center (CSRC) of China is dean of graduate school of Chinese Academy of Engineer- Academy of Engineering Physics (CAEP) in 2020. He is cur- ing Physics (CAEP), and a chair professor at Beijing Com- rently a visiting scholar funded by Bo-Qiang Program at the putational Science Research Center (CSRC). He was selected Graduate School of China Academy of Engineering Physics as an Academician of CAS in 2009, and a Fellow of TWAS (GSCAEP). His-research interests include theoretical study on (the World Academy of Sciences) in 2011. His-research in- quantum thermodynamics and black hole information para- terests include quantum physics, mathematical physics and dox, as well as theoretical and experimental study on ﬁnite- quantum information processing. His-studied focus on the fun- time thermodynamics. He has published more than 10 papers damental aspects of quantum mechanics, e.g., open quan- in Phys. Rev. Lett., Phys. Rev. E, Nucl. Phys. B and other SCI- tum system approaches to quantum measurement and de- indexed journals. These papers have been cited more than 100 coherence, and quantum statistical thermodynamics. In re- times. His-honors include Beijing Outstanding Graduates, Na- cent years, he mainly explored photon transmission in low- tional Scholarship, and Outstanding Students Award of China dimensional structures, quantum coherence eﬀects in biolog- Academy of Engineering Physics. ical processes, and reliability of complex systems, etc. He achieved many prizes including the National Award for Nat- ural Sciences in China, the First Order Prize for Excellence Young Scientist in Chinese Academy of Sciences, the National Prize of Science and Technology for Yang Scientists and Ci- tation Classic Award by SCI et al. He is also honored by the National Model Employee of China.