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Question: Fundamentally, is the existence of negative temperatures a consequence of (a) the violation of entropy postulates, (b) inequilibrium, or (c) finite number of configurations?

Context: In my statistical mechanics class, we first began by claiming the existence of a function $S$, called entropy that contains all information of a (isolated) system (equivalently, the partition function as we move from microcanonical to canonical systems). We postulate several properties of the entropy function:

• Entropy is a concave,
• $\frac{\partial S}{\partial E} > 0$,
• $S$ is positively homogenous of degree 1, i.e.: Entropy is an extensive quantity, as exemplified by $S\left(\lambda E, \lambda X_1, \dots, \lambda X_m \right) = \lambda S\left(E, X_1, \dots, X_m \right),$ where $X_i$ are extensive parameters (thermodynamic quantities).

Then, if the system is in equilibrium, we can define the temperature of the system by $$\frac{1}{T} = \frac{\partial S}{\partial E},$$ where it is implicit that $X_i$ is held constant.

Now, considering the simplest model that yields negative temperatures: $N$ noninteracting two-level particles of fixed positions. It is easy to derive that the entropy $S$ as a function of energy $E$ is a parabola that decreases for $E > \frac{1}{2}\left( E_\text{max} - E_\text{min} \right)$, as seen in the graph here. My first thought was the violation of $\frac{\partial S}{\partial E} > 0$ (and hence the entropy postulate) is a consequence of the finite number of configurations, is the fundamental reason to the existence of negative temperature in this system. However, my tutor has repeatedly spoke of the violation of the entropy postulates as being the fundamental reason (is there circular logic here?), and my lecturer instead stating that negative temperatures are result of systems that are not in equilibrium.

Am I misunderstanding their points?

Remark 1: The finite number of configurations in a thermodynamic system is also mentioned in this wikipedia article here. The following sentence is succinct in describing the thought I had.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy.

Remark 2: In the course of reading various posts on StackEx regarding negative temperatures, I had stumbled onto this, but it is somewhat beyond me, and unsure if it is relevant here.

Question: Fundamentally, is the existence of negative temperatures a consequence of (a) the violation of entropy postulates, (b) inequilibrium, or (c) finite number of configurations?

Context: In my statistical mechanics class, we first began by claiming the existence of a function $S$, called entropy that contains all information of a (isolated) system (equivalently, the partition function as we move from microcanonical to canonical systems). We postulate several properties of the entropy function:

• Entropy is concave,
• $\frac{\partial S}{\partial E} > 0$,
• $S$ is positively homogenous of degree 1, i.e.: Entropy is an extensive quantity, as exemplified by $S\left(\lambda E, \lambda X_1, \dots, \lambda X_m \right) = \lambda S\left(E, X_1, \dots, X_m \right),$ where $X_i$ are extensive parameters (thermodynamic quantities).

Then, if the system is in equilibrium, we can define the temperature of the system by $$\frac{1}{T} = \frac{\partial S}{\partial E},$$ where it is implicit that $X_i$ is held constant.

Now, considering the simplest model that yields negative temperatures: $N$ noninteracting two-level particles of fixed positions. It is easy to derive that the entropy $S$ as a function of energy $E$ is a parabola that decreases for $E > \frac{1}{2}\left( E_\text{max} - E_\text{min} \right)$, as seen in the graph here. My first thought was the violation of $\frac{\partial S}{\partial E} > 0$ (and hence the entropy postulate) is a consequence of the finite number of configurations, is the fundamental reason to the existence of negative temperature in this system. However, my tutor has repeatedly spoke of the violation of the entropy postulates as being the fundamental reason (is there circular logic here?), and my lecturer instead stating that negative temperatures are result of systems that are not in equilibrium.

Am I misunderstanding their points?

Remark 1: The finite number of configurations in a thermodynamic system is also mentioned in this wikipedia article here. The following sentence is succinct in describing the thought I had.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy.

Remark 2: In the course of reading various posts on StackEx regarding negative temperatures, I had stumbled onto this, but it is somewhat beyond me, and unsure if it is relevant here.

Question: Fundamentally, is the existence of negative temperatures a consequence of (a) the violation of entropy postulates, (b) inequilibrium, or (c) finite number of configurations?

Context: In my statistical mechanics class, we first began by claiming the existence of a function $S$, called entropy that contains all information of a (isolated) system (equivalently, the partition function as we move from microcanonical to canonical systems). We postulate several properties of the entropy function:

• Entropy is a concave,
• $\frac{\partial S}{\partial E} > 0$,
• $S$ is positively homogenous of degree 1, i.e.: Entropy is an extensive quantity, as exemplified by $S\left(\lambda E, \lambda X_1, \dots, \lambda X_m \right) = \lambda S\left(E, X_1, \dots, X_m \right),$ where $X_i$ are extensive parameters (thermodynamic quantities).

Then, if the system is in equilibrium, we can define the temperature of the system by $$\frac{1}{T} = \frac{\partial S}{\partial E},$$ where it is implicit that $X_i$ is held constant.

Now, considering the simplest model that yields negative temperatures: $N$ noninteracting two-level particles. It is easy to derive that the entropy $S$ as a function of energy $E$ is a parabola that decreases for $E > \frac{1}{2}\left( E_\text{max} - E_\text{min} \right)$, as seen in the graph here. My first thought was the violation of $\frac{\partial S}{\partial E} > 0$ (and hence the entropy postulate) is a consequence of the finite number of configurations, is the fundamental reason to the existence of negative temperature in this system. However, my tutor has repeatedly spoke of the violation of the entropy postulates as being the fundamental reason (is there circular logic here?), and my lecturer instead stating that negative temperatures are result of systems that are not in equilibrium.

Am I misunderstanding their points?

Remark 1: The finite number of configurations in a thermodynamic system is also mentioned in this wikipedia article here. The following sentence is succinct in describing the thought I had.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy.

Remark 2: In the course of reading various posts on StackEx regarding negative temperatures, I had stumbled onto this, but it is somewhat beyond me, and unsure if it is relevant here.

Question: Fundamentally, is the existence of negative temperatures a consequence of (a) the violation of entropy postulates, (b) inequilibrium, or (c) finite number of configurations?

Context: In my statistical mechanics class, we first began by claiming the existence of a function $S$, called entropy that contains all information of a (isolated) system (equivalently, the partition function as we move from microcanonical to canonical systems). We postulate several properties of the entropy function:

• Entropy is a concave,
• $\frac{\partial S}{\partial E} > 0$,
• $S$ is positively homogenous of degree 1, i.e.: Entropy is an extensive quantity, as exemplified by $S\left(\lambda E, \lambda X_1, \dots, \lambda X_m \right) = \lambda S\left(E, X_1, \dots, X_m \right),$ where $X_i$ are extensive parameters (thermodynamic quantities).

Then, if the system is in equilibrium, we can define the temperature of the system by $$\frac{1}{T} = \frac{\partial S}{\partial E},$$ where it is implicit that $X_i$ is held constant.

Now, considering the simplest model that yields negative temperatures: $N$ noninteracting two-level particles of fixed positions. It is easy to derive that the entropy $S$ as a function of energy $E$ is a parabola that decreases for $E > \frac{1}{2}\left( E_\text{max} - E_\text{min} \right)$, as seen in the graph here. My first thought was the violation of $\frac{\partial S}{\partial E} > 0$ (and hence the entropy postulate) is a consequence of the finite number of configurations, is the fundamental reason to the existence of negative temperature in this system. However, my tutor has repeatedly spoke of the violation of the entropy postulates as being the fundamental reason (is there circular logic here?), and my lecturer instead stating that negative temperatures are result of systems that are not in equilibrium.

Am I misunderstanding their points?

Remark 1: The finite number of configurations in a thermodynamic system is also mentioned in this wikipedia article here. The following sentence is succinct in describing the thought I had.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy.

Remark 2: In the course of reading various posts on StackEx regarding negative temperatures, I had stumbled onto this, but it is somewhat beyond me, and unsure if it is relevant here.

Question: Fundamentally, is the existence of negative temperatures a consequence of (a) the violation of entropy postulates, (b) inequilibrium, or (c) finite number of configurations?

Context: In my statistical mechanics class, we first began by claiming the existence of a function $S$, called entropy that contains all information of a (isolated) system (equivalently, the partition function as we move from microcanonical to canonical systems). We postulate several properties of the entropy function:

• Entropy is a concave,
• $\frac{\partial S}{\partial E} > 0$,
• $S$ is positively homogenous of degree 1, i.e.: Entropy is an extensive quantity, as exemplified by $S\left(\lambda E, \lambda X_1, \dots, \lambda X_m \right) = \lambda S\left(E, X_1, \dots, X_m \right),$ where $X_i$ are extensive parameters (thermodynamic quantities).

Then, if the system is in equilibrium, we can define the temperature of the system by $$\frac{1}{T} = \frac{\partial S}{\partial E},$$ where it is implicit that $X_i$ is held constant.

Now, considering the simplest model that yields negative temperatures: $N$ noninteracting two-level particles. It is easy to derive that the entropy $S$ as a function of energy $E$ is a parabola that decreases for $E > \frac{1}{2}\left( E_\text{max} - E_\text{min} \right)$, as seen in the graph here. My first thought was the violation of $\frac{\partial S}{\partial E} > 0$ (and hence the entropy postulate) is a consequence of the finite number of configurations, is the fundamental reason to the existence of negative temperature in this system. However, my tutor has repeatedly spoke of the violation of the entropy postulates as being the fundamental reason (is there circular logic here?), and my lecturer instead stating that negative temperatures are result of systems that are not in equilibrium.

Am I misunderstanding their points?

Remark 1: The finite number of configurations in a thermodynamic system is also mentioned in this wikipedia article here. The following sentence is succinct in describing the thought I had.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy.

Remark 2: In the course of reading various posts on StackEx regarding negative temperatures, I had stumbled onto this, but it is somewhat beyond me, and unsure if it is relevant here.

Question: Fundamentally, is the existence of negative temperatures a consequence of (a) the violation of entropy postulates, (b) inequilibrium, or (c) finite number of configurations?

Context: In my statistical mechanics class, we first began by claiming the existence of a function $S$, called entropy that contains all information of a (isolated) system (equivalently, the partition function as we move from microcanonical to canonical systems). We postulate several properties of the entropy function:

• Entropy is a concave,
• $\frac{\partial S}{\partial E} > 0$,
• $S$ is positively homogenous of degree 1, i.e.: Entropy is an extensive quantity, as exemplified by $S\left(\lambda E, \lambda X_1, \dots, \lambda X_m \right) = \lambda S\left(E, X_1, \dots, X_m \right),$ where $X_i$ are extensive parameters (thermodynamic quantities).

Then, if the system is in equilibrium, we can define the temperature of the system by $$\frac{1}{T} = \frac{\partial S}{\partial E},$$ where it is implicit that $X_i$ is held constant.

Now, considering the simplest model that yields negative temperatures: $N$ noninteracting two-level particles. It is easy to derive that the entropy $S$ as a function of energy $E$ is a parabola that decreases for $E > \frac{1}{2}\left( E_\text{max} - E_\text{min} \right)$, as seen in the graph here. My first thought was the violation of $\frac{\partial S}{\partial E} > 0$ (and hence the entropy postulate) is a consequence of the finite number of configurations, is the fundamental reason to the existence of negative temperature in this system. However, my tutor has repeatedly spoke of the violation of the entropy postulates as being the fundamental reason (is there circular logic here?), and my lecturer instead stating that negative temperatures are result of systems that are not in equilibrium.

Am I misunderstanding their points?

Remark 1: The finite number of configurations in a thermodynamic system is also mentioned in this wikipedia article here. The following sentence is succinct in describing the thought I had.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy.

Remark 2: In the course of reading various posts on StackEx regarding negative temperatures, I had stumbled onto this, but it is somewhat beyond me, and unsure if it is relevant here.