# If two bodies are completely insulated from each other and are at the same temperature, Are they said to be in thermal equilibrium with each other?

I have read, "Two bodies are said to be in thermal equilibrium if there is no net flow of heat from one to another" And also, "Heat is the form of energy transferred between two systems or a system and it's surroundings by virtue of temperature difference". According to my understanding, when the temperature difference between two bodies is zero, no heat flows between them and they are said to be in thermal equilibrium. Does that mean, Two bodies, completely insulated from each other as well as their surroundings (both are at the same temperature) are in thermal equilibrium with each other? Or is contact between bodies a prerequisite to define equilibrium?

• One aspect of thermal equilibrium is temperatures within the system should be spatially uniform and temporarily constant. These conditions would not necessarily be met when, for example, two ideal gas systems initially thermally and mechanically isolated from one another having equal temperatures but not equal pressures are brought together. Pressure disequilibrium may initially result in the occurrence of non spatially uniform and non temporarily constant temperatures within each system, until mechanical equilibrium is established. Oct 28, 2023 at 17:40

Two objects don't need to be in direct contact to be in thermal equilibrium. It's all about whether there's any net heat flow between them once you brought them together. If there's no heat being exchanged, they're in thermal equilibrium.

This can be completely described by the 0-zero law, which roughly states that

If two systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with one another.

This is the same as saying that there's no heat flow between the tree bodies.

Lastly, let me give you a simple example. Hypothetically, imagine a thermos flask filled with hot coffee. If you pour some into two separate cups, the coffee in each cup (immediately after pouring) is at the same temperature. Even though these two cups of coffee aren't in direct contact with each other, they're in thermal equilibrium because there's no net heat flow between them. Of course, both will exchange heat with their surroundings, but for the moment, they can be considered to be in thermal equilibrium.

I suggest the book Thermodynamics, by Fermi - as far as I remember he discussed very generally the zeroth law there. Or Thermodynamics and an Introduction to Thermostatistics by Callen.

• I guess I would say that, once separated, the two are no longer in equilibrium with each other. To me being in equilibrium implies (no net) energy exchange to ensure equilibrium. Oct 28, 2023 at 15:51
• @JonCuster why not? If they have the same temperature when you bring them back into contact, there won't be any heat exchange. Moreover, imagine two bodies that aren't in contact but share the same environment. They will thermalize and be described by the thermal Gibbs state at the same temperature as the environment.
– Alex
Oct 28, 2023 at 19:33

It would be misleading to say that 2 bodies are in thermal equilibrium with each other if they are completely isolated from each other, whether or not they have the same temperature. After all, the temperature of either body could change by giving off heat to the surrounding, but still there would not be any heat flow from one to the other as they are isolated from each other. You generally can not speak of an 'equilibrium' if there is not some kind of contact between the instances said to be in equilibrium with each other.

• So if you're saying that a composite system, which is locally separated, interacting with a common thermal bath, cannot reach equilibrium? Well, we know that the joint system will thermalize and will be described by a thermal Gibbs state, proportional to the temperature of the environment they are immersed in.
– Alex
Oct 28, 2023 at 19:39
• @Alex If the two objects share a common thermal bath then they are not thermally insulated from each other (which is the condition the OP mentioned). It would be nonsense to say that two objects are in thermal equilibrium with each other when there is no thermal contact, directly or indirectly, possible between the two (take for instance objects on Earth and on Mars that happen to be at the same temperature). Oct 29, 2023 at 14:41
• I might be misunderstanding your comment but thermal equilibrium states are universally described by the Boltzmann-Gibbs distribution. Once you brought two states at the same temperature into contact, there won't be energy flow between them.
– Alex
Oct 30, 2023 at 9:40
• @Alex You can not bring two states into contact if they are supposed to be completely isolated from each other. Besides, it it not true that any equilibrium state is characterized by the Boltzmann-Gibbs/ Maxwell distribution. See my web page plasmaphysics.org.uk/maxwell.htm for more. Oct 30, 2023 at 19:44
• consider two qubits. The first is in state $\rho_A = p_1 |0 \rangle\langle 0| + p_2 |1\rangle \langle 1 |$, the second in $\rho_B = q_1 |0 \rangle\langle 0| + q_2 |1\rangle \langle 1 |$. Both have Hamiltonian $H = E|1\rangle \langle 1|$. Interacting with a thermal bath at inverse temperature $\beta$, their steady state is $\frac{e^{-\beta H}}{Z}$. Couple them via a linear Hamiltonian, and the energy transfer is zero!
– Alex
Oct 30, 2023 at 20:10