How to tell if two system has same temperature?

[KFC]

Two systems are separated by a membrane which allow heat and partcle exchange. Both systems have same volume but different (interaction) potential. Do these two systems have same temperature?

I think they have same temperature because such system could come to equilibrium when time is longer enough, right?

ps. The whole system is isolated

 

[Mapes]

The standard approach here is to write the internal energy of the systems in differential form (e.g.,

$$dU = T_1\,dS_1 +T_2\,dS_2- p_1\,dV_1 - p_2\,dV_2+ \mu_1\,dN_1+ \mu_2\,dN_2+ E_1\,dq_1+ E_2\,dq_2+\dots$$

where E is electrical potential and q is charge), solve for dS, and set this to zero to find what would happen at equilibrium.

I think they have same temperature because such system could come to equilibrium when time is longer enough, right?

Not necessarily; equilibrium doesn't mean that the properties are homogeneous. If energy is coupled to charge (which would be the case for individual particles), then a difference in electrical potential between the two systems could lead to a temperature gradient, as exemplified by the thermoelectric effect.

 

[KFC]

The standard approach here is to write the internal energy of the systems in differential form (e.g.,

$$dU = T_1\,dS_1 +T_2\,dS_2- p_1\,dV_1 - p_2\,dV_2+ \mu_1\,dN_1+ \mu_2\,dN_2+ E_1\,dq_1+ E_2\,dq_2+\dots$$

where E is electrical potential and q is charge), solve for dS, and set this to zero to find what would happen at equilibrium.


Not necessarily; equilibrium doesn't mean that the properties are homogeneous. If energy is coupled to charge (which would be the case for individual particles), then a difference in electrical potential between the two systems could lead to a temperature gradient, as exemplified by the thermoelectric effect.

I fogot to say, no interaction b/w particles