2.1. Postulates and Definitions of State
In addition to empirical facts and physical laws, the TCI’s interpretation of the physical state is based on the following postulates and definitions:
Postulate one. The zeroth law of thermodynamics establishes that the temperature of a thermally equilibrated system is a measurable property.
Postulate two. The third law of thermodynamics establishes that absolute zero can be approached but never be attained.
Postulate three. There are no unobservable “hidden” variables. The physical properties of state are measurable and perfect measurement completely describes a system’s physical state.
Definition 1. A system’s ambient temperature, Ta, equals the positive temperature of the system’s actual surroundings with which it interacts or potentially interacts.
Definition 2. A system’s total energy, E, equals the system’s potential work as measured on the surroundings in the limit of absolute zero.
Definition 3. A system’s exergy, X, is defined by its potential work as measured at the ambient surroundings.
Definition 4. A system is in its ground state when its temperature equals the ambient temperature and its exergy equals zero. The system’s ground state is uniquely defined by its equilibrium with its ambient surroundings.
Definition 5. A system’s ground state energy, Qgs, is the ambient ground state’s potential work capacity as measured at the limit of absolute zero.
Definition 6. A system’s energy is defined by Esys = E − Qgs.
Definition 7. A system’s ambient heat is defined by Q = Esys − X.
Definition 8. Perfect measurement of a state is a reversible and open system process of transition from a system’s initial state to its ground state.
Definition 9. A system’s entropy is defined by S = Q/Ta.
Postulate four (second law of thermodynamics). An irreversible process dissipates exergy into ambient heat. For irreversible change within an isolated system at constant Ta, ΔX < 0.
The TCI is a conceptual model and a simplification of reality. It is based on empirical facts and empirically justified assumptions that provide the logical foundation for the TCI and its explanations of empirical facts within its domain of empirical validation.
Postulate one establishes temperature as a measurable property. The zeroth law of thermodynamics defines a system’s temperature by the measurable temperature of a thermometer or probe with which it is thermally equilibrated.
Postulate two says that absolute zero can be approached but never attained. No system is perfectly isolated from its surroundings and all systems have a positive ambient temperature. Even the universe, which by definition has no physical surroundings, has an ambient energy background for the exchange of photons, defined by its cosmic microwave temperature at 2.7 kelvins.
Postulate three is a statement about the TCI’s interpretation of physical reality. Postulate three defines physical reality by perfect measurement. The microstate, which expresses everything measurable and knowable about a system, is therefore a complete description of the physical state. “State”, without any qualification, will refer both to a system’s measurable microstate and to its underlying physical state.
Definition one defines a system’s ambient temperature by the temperature of the system’s ambient surroundings, whether or not the system is thermally equilibrated with its surroundings. Definition two follows the Hamiltonian conceptual framework by defining a system’s total energy with respect to absolute zero. However, whereas the HCF only considers total energy the TCI resolves total energy into thermocontextual components (definitions three and five–seven), given by:
The system’s energy (Esys), exergy (X), and ambient heat (Q) are all thermocontextual properties of state, measured and defined with respect to the system’s ambient ground state (definition four), which defines their zero values. The ambient ground state energy (Qgs) is defined by measurements in the limit of absolute zero.
The TCI’s ambient ground state energy is a generalization of quantum mechanics’ absolute ground state energy. Whereas the ambient ground state is in equilibrium with a system’s actual ambient surroundings at a positive temperature, quantum mechanics’ absolute ground state is defined at absolute zero. However, even in the limit of absolute zero, measurements reveal a positive ground state energy due to Heisenberg’s uncertainty principle and the observed randomness of a particle’s position and momentum.
The ambient temperature, not the system temperature, defines a system’s temperature of thermalization. If a system’s energy is fully thermalized at the ambient temperature, then it has no potential for work and it defines ambient heat.
If a system’s ambient temperature is lower than its temperature, then the system’s internal energy (heat) is only partially thermalized and it has positive thermal exergy. The thermal exergy and ambient heat contents of an increment of heat (
dq) at temperature
T >
Ta are empirically given by:
Thermal exergy is the maximum work that can be derived from heat and an ambient heat sink using a perfectly efficient heat engine.
A system’s total exergy is the sum of its thermal exergy plus the nonthermal kinetic and potential energies of the system’s measurable components. A system’s measurable components generally are not elementary particles and can contain internal exergy (e.g., chemical or nuclear potential energy) in addition to the potential energy resulting from interactions with the particles’ external fields.
When combined with the law of conservation for energy, we can rewrite Equation (1) as:
If the ambient surroundings are fixed then equals zero, and Equation (3) shows that the dissipation of exergy is offset by an increase in ambient heat. Equation (3) also expresses the conservation of energy during changes in the ambient surroundings. A decline in the temperature of the ambient surroundings shifts ground state energy into system energy, and from Equation (2), this shifts ambient heat into exergy, but the isolated system’s total energy is conserved.
Definition eight defines perfect measurement of a system’s state as a reversible open system transition from an initial state into its ambient ground state. Before any interaction with its cooler ambient surroundings, a hot gas has positive exergy and ambient heat. In the limit of reversibility, we can cool the gas via a heat engine until it reaches its ambient ground state, and we can store the gas’s thermal exergy without dissipation or loss (
Figure 1). The process thereby reversibly transitions the system into its ground state with zero system energy. Using the exergy stored in the surroundings, we can reverse the measurement process by reversibly pumping ambient heat back into the gas, restoring the gas to its original state.
Perfect measurement is defined as a reversible open system process, but reversible measurement is not always possible. The quantum watchdog effect (a continuous version of the Zeno effect) shows that a continuously measured (and measurable) state cannot change irreversibly [
17]. The contrapositive of this is equally true: an irreversibly changing system is not continuously measurable. During radioactive decay, for example, exergy is irreversibly dissipated and the particle is not reversibly measurable. The particle therefore does not exist as a TCI state during transition. A system is reversibly measurable only
between irreversible transitions, while it exists as a metastable state across an interval of time.
The TCI includes thermodynamic properties, but there is an important and fundamental difference between the thermocontextual state and the thermodynamic state. The TCI defines a state with respect to an ambient reference state that is fully thermalized at the system’s ambient temperature. Nonequilibrium thermodynamics, in contrast, assumes a non-isothermal system is thermalized at the system’s variable local temperature. There is no well-defined ambient temperature of thermalization, and it does not describe a state within the TCI. The thermodynamic description of an isothermal system, however, does describe the special case TCI state, in which the ambient temperature is defined by the system’s temperature.
2.2. Entropy and Refinement
Definition nine defines entropy by
S =
Q/Ta. As with ambient heat, the TCI entropy is a thermocontextual property of state that is defined relative to the system’s equilibrium ground state at the ambient temperature. The TCI resolves total entropy (diagonal vector in
Figure 2) into two components: the ambient entropy (
Samb, horizontal vector) and the entropy of refinement (
Sref, vertical vector).
Refinement was originally described by the consistent histories interpretation of quantum mechanics [
14] as a consequence of a change in the measurement framework. The TCI extends this idea and defines the entropy of thermal refinement as a consequence of a decline in ambient temperature from an initial
Tai to a final
Taf. The entropy of thermal refinement simply reflects a change in the ambient entropy due to a shift in the temperature scale for measuring entropy. If the ambient temperature is constant (
Tai =
Taf) then
Sref (vertical vector) equals zero and the ambient entropy and TCI entropy are equal.
The entropy of thermal refinement is defined by the integral:
The differential dQ is the incremental change in the system’s ambient heat, thermalized at the fixed temperature Taf, and dq is the incremental change in heat thermalized at temperature T. The first equality is based on definition nine and the second equality follows from Equation (2), with Tai = T and Taf = Ta. In both cases the changes are in response to the change in the ambient temperature prior to any heat transfer or other adjustments to the changed surroundings.
For a system initially thermalized at its system temperature (
Tai =
Tsys), as
Taf and the temperature of re-thermalization approach absolute zero, the ambient heat and the ambient entropy also approach zero. In the limit of
Ta = 0 kelvins the ambient entropy (horizontal vector) is zero. The TCI entropy therefore equals the entropy of refinement, and we get:
In the limit of an ambient temperature of absolute zero, Sref and STCI are both equal to thermodynamics’ third law entropy, defined by the integral term. Conversely, the thermocontextual TCI entropy is a generalization of thermodynamics’ third law entropy, defined for the idealized special case of an ambient temperature of absolute zero.
Refinement is about more than just a decline in ambient temperature: it is a consequence of any decline in a system’s equilibrium ground state energy. The conservation of energy means that a decline in a system’s ground state energy results in an increase in its system energy. Refinement also does more than just increase a system’s ambient heat and entropy: it also increases the system’s exergy. A system can be prepared in equilibrium with its environment of preparation with zero exergy and zero entropy, but when it is cast into lower-energy surroundings it will attain positive entropy and positive exergy.
2.3. Classical and Quantum States
A system’s TCI state is defined by its total energy with respect to absolute zero, by its ambient ground state, and by perfect measurement (definition eight,
Figure 1).
Table 1 describes the TCI interpretations of an ideal classical gas and a hydrogen atom. We initially consider the gas and hydrogen atom to be in equilibrium with their ambient surroundings. We assume that the gas is in equilibrium with a thermal bath at
T = 500 K and fixed pressure and that the hydrogen atom is in equilibrium with a black body at
T = 6000 K. This is below the hydrogen atom’s ionization temperature but hot enough that its single electron is distributed among multiple energy levels, and it is described by a superposed wavefunction.
The n-particle gas’s equilibrium state is defined by its equilibrium temperature and pressure (
Table 1, rows one and two). The equilibrium gas is a special case TCI state with
Ta =
T. The gas’s total energy is defined by the equilibrium temperature and pressure, and its thermocontextual properties all equal zero. The equilibrium gas defines the gas’s “e-contextual” state, where the TCI defines an e-contextual state by equilibrium with its ground state.
The TCI generalizes the gas’s e-contextual state by defining the gas with respect to the system’s actual ambient surroundings. If we start with an equilibrium gas and lower its ambient temperature from 500 K to 300 K the e-contextual state is unchanged, but the gas’s thermocontextual state immediately changes, as shown in
Table 1 (bottom five rows). In particular, prior to any interactions with the new surroundings at 300 K the gas at 500 K has a positive exergy and positive entropy of refinement with respect to its new ground state.
Quantum mechanics defines the hydrogen atom’s quantum state by its superposed wavefunction (
Table 1, row one). The individual eigenfunctions are independent of temperature, but their complex weighting coefficients, and therefore the wavefunction, are functions of the hydrogen atom’s equilibrium temperature. The wavefunction and its equilibrium temperature uniquely specify the atom’s energy expectation value, defined in row two. The superposed wavefunction completely specifies the atom as it exists in equilibrium with its ambient surroundings in its e-contextual state.
If we perfectly isolate the hydrogen atom and lower its ambient temperature to 300 K its wavefunction and e-contextual state are unchanged. Its thermocontextual properties (bottom five rows of
Table 1) do change, however. The thermocontextual state with respect to 300 K describes the atom with positive exergy and entropy. It is a generalization of its special case e-contextual microstate.
If we lower the ambient temperature all the way to absolute zero, there is no possibility of dissipation, randomness, or irreversibility. The TCI describes a property or state that is defined with respect to absolute zero as a “z-contextual” property or state. Ambient heat and entropy are identically zero, and the system energy is equal to its exergy. The deterministic and reversible z-contextual state is an idealized special case of thermocontextuality, with ambient temperature equal to absolute zero. This applies to both classical and quantum states.