We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between the Gibbs entropy φ_(V,N)(E) as a function of mean internal energy E and Massieu's free entropy Ψ_(V,N)(β) as a ...We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between the Gibbs entropy φ_(V,N)(E) as a function of mean internal energy E and Massieu's free entropy Ψ_(V,N)(β) as a function of inverse temperature β. The duality in terms of Legendre–Fenchel transform tells us that Gibbs' thermodynamic entropy is to the law of large numbers(LLN) for arithmetic sample mean values what Shannon's information entropy is to the LLN for empirical counting frequencies in independent and identically distributed data. Proceeding with the same mathematical logic, we identify the energy of the state {ui} as the conjugate variable to the counting of statistical occurrence {mi} and find a Hamilton–Jacobi equation for the Shannon entropy analogous to an equation of state in thermodynamics. An eigenvalue problem that is reminiscent of certain features in quantum mechanics arises in the entropy theory of statistical counting frequencies of Markov correlated data.展开更多
Equations(2)and(6)and the corresponding discussion in the paper[Chin.Phys.Lett.42,056301(2025)]have been corrected.These modiffcations do not affect the results derived in the paper.
In order to avoid the worsening of wealth inequality,it is necessary to explore the influencing factors of wealth distribution and discuss measures to reduce wealth inequality.We investigate the wealth distribution in...In order to avoid the worsening of wealth inequality,it is necessary to explore the influencing factors of wealth distribution and discuss measures to reduce wealth inequality.We investigate the wealth distribution in the goods exchange market by using the kinetic theory of rarefied gas.The trading objects are two kinds of commodities(commodities A and B)and the trading subjects are agents of two groups(dealers and speculators).We deduce the interaction rules according to the principle of utility maximization and consider the transfer of agents in the Boltzmann equation.The steady solution of the Fokker-Planck equation for a special case is obtained and the effects of trading strategy and transfer frequency on the steady distribution are analyzed in numerical experiments.The conclusions illustrate that the transfer of agents is conducive to reducing the inequality of wealth distribution.展开更多
Isobaric molar heat capacity affected by pressures for non-ideal gases is calculated theoretically at specified temperatures by means of gaseous equations of state,i.e.Redlish-Kwong(RK)Equation,SoaveRedlich-Kwong(SRK)...Isobaric molar heat capacity affected by pressures for non-ideal gases is calculated theoretically at specified temperatures by means of gaseous equations of state,i.e.Redlish-Kwong(RK)Equation,SoaveRedlich-Kwong(SRK)Equation,Peng-Robinson(PR)Equation,Virial Equation,coupled with Romberg numeric integral via solving the key obstacle(δV/δT)_(p),and integral(δ^(2)V/δT^(2))_(p).As an example,methane's C_(p)is calculated at constant 300 K but 1 MPa&10 MPa.The calculation results show that less than 2%relative errors occur in comparison with literature values at any specified temperatures and pressures if no phase change survives at elevated pressure P_(2)and temperature T,or when specified temperatures are greater than critical temperatures in spite of elevated pressures.However,greater errors would be present if gases were considered to be ideal,or if temperatures are lower than critical temperatures at elevated pressures(>10 MPa),because C_(p)is the function of both temperature and pressure.In particular,elevated pressures have significant effect on C_(p).展开更多
A thermodynamically complete multi-phase equation of state(EOS)applicable to both dense and porous metals at wide ranges of temperature and pressure is constructed.A standard three-term decomposition of the Helmholtz ...A thermodynamically complete multi-phase equation of state(EOS)applicable to both dense and porous metals at wide ranges of temperature and pressure is constructed.A standard three-term decomposition of the Helmholtz free energy as a function of specific volume and temperature is presented,where the cold component models both compression and expansion states,the thermal ion component introduces the Debye approximation and melting entropy,and the thermal electron component employs the Thomas-Fermi-Kirzhnits(TFK)model.The porosity of materials is considered by introducing the dynamic porosity coefficientαand the constitutive P-αrelation,connecting the thermodynamic properties between dense and porous systems,allowing for an accurate description of the volume decrease caused by void collapse while maintaining the quasi-static thermodynamic properties of porous systems identical to the dense ones.These models enable the EOS applicable and robust at wide ranges of temperature,pressure and porosity.A systematic evaluation of the new EOS is conducted with aluminum(Al)as an example.300 K isotherm,shock Hugoniot,as well as melting curves of both dense and porous Al are calculated,which shows great agreements with experimental data and validates the effectiveness of the models and the accuracy of parameterizations.Notably,it is for the first time Hugoniot P-σcurves up to 10~6 GPa and shock melting behaviors of porous Al are derived from analytical EOS models,which predict much lower compression limit and shock melting temperatures than those of dense Al.展开更多
文摘We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between the Gibbs entropy φ_(V,N)(E) as a function of mean internal energy E and Massieu's free entropy Ψ_(V,N)(β) as a function of inverse temperature β. The duality in terms of Legendre–Fenchel transform tells us that Gibbs' thermodynamic entropy is to the law of large numbers(LLN) for arithmetic sample mean values what Shannon's information entropy is to the LLN for empirical counting frequencies in independent and identically distributed data. Proceeding with the same mathematical logic, we identify the energy of the state {ui} as the conjugate variable to the counting of statistical occurrence {mi} and find a Hamilton–Jacobi equation for the Shannon entropy analogous to an equation of state in thermodynamics. An eigenvalue problem that is reminiscent of certain features in quantum mechanics arises in the entropy theory of statistical counting frequencies of Markov correlated data.
文摘Equations(2)and(6)and the corresponding discussion in the paper[Chin.Phys.Lett.42,056301(2025)]have been corrected.These modiffcations do not affect the results derived in the paper.
文摘In order to avoid the worsening of wealth inequality,it is necessary to explore the influencing factors of wealth distribution and discuss measures to reduce wealth inequality.We investigate the wealth distribution in the goods exchange market by using the kinetic theory of rarefied gas.The trading objects are two kinds of commodities(commodities A and B)and the trading subjects are agents of two groups(dealers and speculators).We deduce the interaction rules according to the principle of utility maximization and consider the transfer of agents in the Boltzmann equation.The steady solution of the Fokker-Planck equation for a special case is obtained and the effects of trading strategy and transfer frequency on the steady distribution are analyzed in numerical experiments.The conclusions illustrate that the transfer of agents is conducive to reducing the inequality of wealth distribution.
文摘Isobaric molar heat capacity affected by pressures for non-ideal gases is calculated theoretically at specified temperatures by means of gaseous equations of state,i.e.Redlish-Kwong(RK)Equation,SoaveRedlich-Kwong(SRK)Equation,Peng-Robinson(PR)Equation,Virial Equation,coupled with Romberg numeric integral via solving the key obstacle(δV/δT)_(p),and integral(δ^(2)V/δT^(2))_(p).As an example,methane's C_(p)is calculated at constant 300 K but 1 MPa&10 MPa.The calculation results show that less than 2%relative errors occur in comparison with literature values at any specified temperatures and pressures if no phase change survives at elevated pressure P_(2)and temperature T,or when specified temperatures are greater than critical temperatures in spite of elevated pressures.However,greater errors would be present if gases were considered to be ideal,or if temperatures are lower than critical temperatures at elevated pressures(>10 MPa),because C_(p)is the function of both temperature and pressure.In particular,elevated pressures have significant effect on C_(p).
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12205023,U2230401,12374056,U23A20537,11904027)。
文摘A thermodynamically complete multi-phase equation of state(EOS)applicable to both dense and porous metals at wide ranges of temperature and pressure is constructed.A standard three-term decomposition of the Helmholtz free energy as a function of specific volume and temperature is presented,where the cold component models both compression and expansion states,the thermal ion component introduces the Debye approximation and melting entropy,and the thermal electron component employs the Thomas-Fermi-Kirzhnits(TFK)model.The porosity of materials is considered by introducing the dynamic porosity coefficientαand the constitutive P-αrelation,connecting the thermodynamic properties between dense and porous systems,allowing for an accurate description of the volume decrease caused by void collapse while maintaining the quasi-static thermodynamic properties of porous systems identical to the dense ones.These models enable the EOS applicable and robust at wide ranges of temperature,pressure and porosity.A systematic evaluation of the new EOS is conducted with aluminum(Al)as an example.300 K isotherm,shock Hugoniot,as well as melting curves of both dense and porous Al are calculated,which shows great agreements with experimental data and validates the effectiveness of the models and the accuracy of parameterizations.Notably,it is for the first time Hugoniot P-σcurves up to 10~6 GPa and shock melting behaviors of porous Al are derived from analytical EOS models,which predict much lower compression limit and shock melting temperatures than those of dense Al.
