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chimay

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- Question about a mathematical detail in the proof of the thermodynamic stability relation

I am watching Kardar's Statistical Mechanics course in my spare time and I am struggling to understand a mathematical detail in the proof of the thermodynamic stability condition. See Eq. I.62 here.

The author considers a homogeneous system at equilibrium with intensive and extensive variables [itex](T,J,\mu)[/itex] and [itex](E,x,N)[/itex], respectively. Then, He imagines that the latter system is divided into two subsystems (A and B) that can exhange energy. Under the assumption that [itex]E,x[/itex] and [itex]N[/itex] are conserved, we have [itex]\delta E_A=-\delta E_B [/itex], [itex]\delta x_A=-\delta x_B [/itex] and [itex]\delta N_A=-\delta N_B [/itex]. What is not clear to me is the statement "Since the intensive variables are themselves functions of the extensive coordinates, to first order in the variations of [itex](E, x, N)[/itex], we have [itex]\delta T_A=-\delta T_B [/itex], [itex]\delta J_A=-\delta J_B [/itex] and [itex]\delta \mu_A=-\delta \mu_B [/itex]."

Can anyone explain to me the previous statement more in detail?

Thank you!

The author considers a homogeneous system at equilibrium with intensive and extensive variables [itex](T,J,\mu)[/itex] and [itex](E,x,N)[/itex], respectively. Then, He imagines that the latter system is divided into two subsystems (A and B) that can exhange energy. Under the assumption that [itex]E,x[/itex] and [itex]N[/itex] are conserved, we have [itex]\delta E_A=-\delta E_B [/itex], [itex]\delta x_A=-\delta x_B [/itex] and [itex]\delta N_A=-\delta N_B [/itex]. What is not clear to me is the statement "Since the intensive variables are themselves functions of the extensive coordinates, to first order in the variations of [itex](E, x, N)[/itex], we have [itex]\delta T_A=-\delta T_B [/itex], [itex]\delta J_A=-\delta J_B [/itex] and [itex]\delta \mu_A=-\delta \mu_B [/itex]."

Can anyone explain to me the previous statement more in detail?

Thank you!