^{1}

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The irreversible mechanism of heat engines is studied in terms of
*thermodynamic consistency* and thermomechanical dynamics (TMD) which is proposed for a method to study nonequilibrium irreversible thermodynamic systems. As an example, a water drinking bird (DB) known as one of the heat engines is specifically examined. The DB system suffices a rigorous experimental device for the theory of nonequilibrium irreversible thermodynamics. The DB nonlinear equation of motion proves explicitly that nonlinear differential equations with time-dependent coefficients must be classified as independent equations different from those of constant coefficients. The solutions of nonlinear differential equations with time-dependent coefficients can express emergent phenomena: nonequilibrium irreversible states. The
*couplings* among mechanics, thermodynamics and time-evolution to nonequilibrium irreversible state are defined when the internal energy, thermodynamic work, temperature and entropy are integrated as a spontaneous thermodynamic process in the DB system. The physical meanings of the time-dependent entropy,
*T*(
*t*)d
*S*(
*t*), , internal energy, d
*Ɛ*(
*t*), and thermodynamic work, dW(
*t*), are defined by the progress of time-dependent Gibbs relation to thermodynamic equilibrium. The thermomechanical dynamics (TMD) approach constitutes a method for the nonequilibrium irreversible thermodynamics and transport processes.

A water drinking bird (DB) is a delightful, scientific toy that produces a simple back and forth motion by occasionally dipping its beak in water as if it keeps moving only with water supply. On the contrary to the delicate and slender figure, the DB system is an excellent educational tool for mechanical and thermodynamic transport systems of heat and energy as Albert Einstein and many other scientists have been fascinated [

The mathematical definition of thermodynamic equilibrium is given by the fundamental relation in thermodynamics [

∂ S ( E , V ) ∂ E | V , N , ⋯ = 1 T , (1.1)

with the entropy S , internal energy E , temperature T, at fixed volume V and particle number N, ⋯ , at thermodynamic equilibrium (Boltzmann constant, k B ≡ 1 ). The first and the second laws of thermodynamics and relations among thermodynamic functions are incorporated in Equation (1.1) at thermal equilibrium. The drinking bird begins with a simple pendulum motion of classical mechanics; gas in head-bulb and volatile water in a glass tube are at thermal equilibrium but gradually evolve to nonequilibrium irreversible states and then, the mechanism leads to a drinking motion. One may think that motion of the drinking bird is easy and simple, but one should be very careful that the drinking bird’s motion develops from mechanical and thermodynamic equilibriums to a nonequilibrium irreversible state, which is one of the fundamental topics for the research and the laws of physics.

We employ the fundamental relation (1.1) for the time-evolution from mechanical and thermodynamic equilibriums to a nonequilibrium irreversible state and vice versa. The change of the thermal state is related to time-evolution of the internal energy, E ( t ) , entropy S ( t ) , temperature T ( t ) and work W ( t ) in the DB system, and the work W ( t ) includes a time-change of volume V ( t ) . Hence, we suppose the transformation from mechanical and thermodynamic states to a nonequilibrium irreversible state (NIS) by:

T ∂ S ( E , V ) ∂ E | V , N , ⋯ = 1 | t ≤ t c → t c < t T ( t ) ∂ S ( t , E ( t ) , W ( t ) ) ∂ E ( t , W ( t ) ) | t ≤ t d , (1.2)

where the time t c is a “critical time” (onset time) for the transition, and t d is the corresponding drinking (dipping) time. The transition from a NIS to an equilibrium state after dipping is similarly investigated by:

T ( t ) ∂ S ( t , E ( t ) , W ( t ) ) ∂ E ( t , W ( t ) ) | t ≤ t d → t d < t T ∂ S ( E , V ) ∂ E | V , N , ⋯ = 1 . (1.3)

Though detailed heat conduction mechanisms should be introduced for transitions at t c and t d shown by arrows in (1.2) and (1.3), the transitions are mathematically simplified and expressed by using piecewise continuous function (the step function) in the thermomechanical model [

The transition from equilibrium to a NIS starts at t c shown by (1.2), and a different mode of oscillation driven by thermodynamic work (up-going volatile water in the glass tube) is observed until a DB’s dipping. After a DB’s dipping, the volatile water in the glass tube is artificially returned to the lower bulb by a mechanical trick. The mathematical technicality corresponding to the mechanical trick is produced with the step function and termed as “initialization” [

The time period, t c < t < t d , characterizes a NIS, and one can check different modes of oscillations (slow and elongated, slightly-increasing oscillations) developing to water drinking motion, and after drinking, the DB motion is decoupled to an equilibrium, which is composed of independent mechanical and thermodynamic states. The time-evolution from equilibrium to a NIS and then, to an equilibrium state can be clearly observed from the change of oscillation modes. In other words, the emergent coupling and decoupling of mechanical and thermodynamic states can be numerically measured in DB’s oscillations. Although the nonlinear differential equation with constant coefficients is used as the DB’s starting equation of motion by way of Hamiltonian (or Lagrangian) method, the DB motion cannot be reproduced by the nonlinear differential equation with constant coefficients. The mathematical equation of motion for the water drinking motion is a nonlinear differential equation with time-dependent coefficients, which has independent solutions to those of nonlinear differential equation with constant coefficients. The nonlinear differential equations with constant or time-dependent coefficients must be categorized as different classes of nonlinear differential equations, which is a mathematically important property discovered in the DB analysis, which will be explained in Section 2 and Section 3.

The macroscopic (total or global) entropy, d S and microscopic (constituent or local) entropies d S i have been often discussed in thermodynamic equilibrium, or linear and near-equilibrium states so that the entropy relation, d S = ∑ i d S i , holds. One should realize that fundamental complications will arise when the internal energy, work, heat, particle flows of photon, electron, nuclear particles, ..., are self-consistently coupled to each other in nonequilibrium and irreversible processes, resulting in d S ≠ ∑ i d S i . This would be a reason why one needs to restrict thermodynamic variables for adequate descriptions of nonequilibrium systems [

The terms, entropy, entropy production or dissipation, are known to be defined and used by different researchers in different ways, such as the maximum or minimum entropy production, minimum energy dissipation, etc., which is creating much confusion and logically circular arguments and statements [

Modern technologies are progressing towards higher speed, power, miniaturization, efficient energy use and minimization of waste of energy, necessary for supporting ecological diversity and energy sustainable societies. In addition, space technologies will demand high energy use and energy efficiency. The issues in diverse fields of different energy conversions, harvesting and storage technologies are important for modern society to solve existing environmental problems [

The analysis of drinking bird made us study the fundamental structure for theories of science, which would also help people in science reflect on physical principles of sciences; it is summarized that the theory of science should be examined in terms of reproducibility, self-consistency and testability. These criteria are difficult to achieve in general for scientific theories and models, but it would be reasonable to say that the four fundamental mechanics (Analytical mechanics, Thermodynamics, Electromagnetism, Quantum mechanics) and associated models, such as special relativity, quantum electrodynamics (QED) seem to maintain, and they have shown reproducibility, theoretical self-consistency and testability, resulting in great contributions to modern technologies. The DB papers [

The time-dependent internal energy E ( t ) , nonlinear differential equation of motion with time-dependent coefficients, coupled and decoupled oscillations are discussed with a short review of the thermomechanical DB model in Section 2. Then, the time-evolution of internal energy E ( t ) , work W ( t ) , total entropy S ( t ) and temperature T ( t ) , thermodynamic consistency of the DB system and heat engines are examined in Section 3. The foundation of thermomechanical dynamics (TMD) for nonequilibrium irreversible thermodynamics is discussed in Section 4. The conclusions and perspectives of scientific ways of thinking are discussed in Section 5.

A drinking bird and its topological deformation and the angle θ ( t ) are shown in

E = 1 2 I 0 θ ˙ 2 − { m 2 g a − m 1 g ( b − l 2 ) − m 3 g b } cos θ , (2.1)

where the DB’s length is l = a + b . The moment of inertia is given by the sum, I 0 = I 1 + I 2 + I 3 , the moment of inertia of head I 3 , glass tube I 1 , and lower bulb I 2 , respectively [

I 1 = m 1 l 2 ( 1 3 − ( 1 − b l ) + ( 1 − b l ) 2 ) . (2.2)

The effective mass m ∗ is introduced for convenience as,

m ∗ = m 2 a l − m 1 ( b l − 1 2 ) − m 3 b l , (2.3)

and the DB energy is written with m ∗ as,

E = 1 2 I 0 θ ˙ 2 − l g m ∗ cos θ , (2.4)

where the potential energy is measured from the axis of rotation in

θ ¨ + c θ ˙ + g l m ∗ I 0 sin θ = 0 . (2.5)

The DB motion derived from (2.4) and (2.5) can express, for example, back-and-forth solutions around θ = 0 , or up-side-down solutions around θ = π , as shown respectively in

θ n = ± n π ( n = 0 , 1 , 2 , 3 , ⋯ ) , and the solution suddenly changes from the θ = 0

converging solution to the others when certain parameter values of mass m and moment of inertia I 0 are given (other parameters, g , l , c , are fixed). The property of splitting into independent solutions is known as bifurcation phenomena for nonlinear differential equations with constant coefficients [

The up-going liquid in a glass tube changes the DB moment of inertia and internal energy, and we assumed a constant velocity, v ( t ) = v 0 , for the up-going liquid [

E ( t ) = 1 2 I ( t ) θ ˙ 2 − l g m ∗ ( t ) cos θ , (2.6)

and detailed expressions of I ( t ) and m ∗ ( t ) are discussed in the paper [

θ ¨ + c θ ˙ + g l m ∗ ( t ) I ( t ) sin θ = 0 . (2.7)

The nonlinear differential Equations (2.6) and (2.7) should be compared with the energy and mechanical equation of motion (2.4) and (2.5), and numerical solutions are respectively shown in

One should also note that the equation of motion (2.7) cannot be derived from (2.6) by way of a mechanical, Lagrangian or Hamiltonian method, because the energy (2.6) is explicitly time-dependent and equivalently non-conservative. However, the physical results and mechanism of NDE with time-dependent coefficients, the DB’s time-dependent m ∗ ( t ) and I ( t ) , driven by time-dependent energy E ( t ) and thermodynamic work W ( t ) are consistent with DB’s motion as a NIS. The arrow in

( x 2 ( t ) , m ∗ ( t ) , I ( t ) ) → ( a , m ∗ , I 0 ) t d , (2.8)

at a drinking time t d . This is a mathematical simplification, which must be extended by including mechanism of heat conduction, diffusion and ignition phenomina [

The motion from the initial oscillations to the first critical time t c ∼ 24.0 ( s ) , then to the dipping at t d ∼ 38.8 ( s ) and oscillations after t d is shown in

a critical time t 1 c ≃ 24.0 ( s ) , and the change of oscillation-mode induced by thermodynamically-driven forced oscillations is observed from t 1 c to t 1 d ∼ 38.8 ( s ) . The critical time t c signifies the onset and transition from equilibrium to a NIS and the reversed transition occurs at the dipping time t d . Similarly, the DB’s 1st and 2nd oscillations are shown in

The mechanical energy conservation law loses its meaning, but instead, the total conservation of heat and energy is restored in the NIS. The physical coefficients of the internal energy and equation of motion become time-dependent and progress to the NIS, which is fundamentally different from mechanical and thermal equilibriums. These phenomena are not supposed to be obtained from the continuous unitary-transformation in time in classical and quantum mechanical systems because time-symmetry is broken in a NIS. The time-dependence of thermodynamic work W ( t ) is considered to be spontaneously produced and never infinitesimally slowly constructed in a NIS, and thermodynamic work also depends on levels of technology to convert the thermal energy. The physical mechanism to induce time-dependent coefficients of energy and the nonlinear equation of motion are essential points to be scrutinized by the concept of thermodynamic work, internal energy and entropy in a NIS.

The time variations of the moment of inertia, I ( t ) , corresponding to

The concept of coupled or decoupled mechanical and thermodynamic systems is important to study mechanism of time evolutions from a NIS to thermal and mechanical equilibriums and vice versa. In case that heat or entropy flows could not produce sufficient work to affect each other, the NIS proceeds to respective equilibriums, which can be experimentally checked and may correspond to linear thermodynamics or not far from thermodynamic equilibrium. Therefore, the DB system can be in a state of thermodynamic equilibrium with the oscillation around θ = 0 and small differences of heat and energy as near thermodynamic equilibrium, expressed as T ∂ S ( E , V ) / ∂ E | V , N , ⋯ ≈ 1 . With the condition, the linear thermodynamics and flux equations can be discussed in the DB analysis. An example of numerical simulations of fundamental relations, (1.1)-(1.3), is specifically examined in the next section.

The fundamental macroscopic relations among the internal energy, work, heat and entropy, particle and other physical flows, are intertwined and become self-consistent interacting quantities, and so, one should be careful to study and construct overall structural views in NISs. It becomes difficult to select macroscopic and microscopic variables, because constituent physical quantities do not maintain, for instance, δ S = ∑ i δ S i . The current irreversible thermomechanical DB model exhibits “global” motion caused by changes of internal energy, entropy flow and “local” change of thermodynamic work by liquid movement in the glass tube and can produce the DB dynamics reasonably well. One could include the gas-phase, liquid phase and gas-liquid interface many-body interactions, local internal energies and entropies of respective phases; however, the inclusion of a local constituent variable will produce extremely complicated interactions among constituents, which makes difficult to reproduce even a DB’s simple back-and-forth motion.

The simple and approximate explanations to DB’s motion employ pressure, temperature and volume in gas phase (Boyle-Charles’ law, or the ideal gas law) to explain up-going water in a glass tube, though a DB system is not at thermodynamic equilibrium. Moreover, the concept of local equilibrium is not assumed in the thermomechanical DB model; however, the local equilibrium or linearity can be reproduced with the conditions of small entropy flow and T ∂ S ( E , V ) / ∂ E | V , N , ⋯ ≈ 1 . One should realize that the current approach is not restricted to near equilibrium. Thus, the thermomechanical model of a NIS with time-dependent thermodynamic functions, ( E ( t ) , T ( t ) S ( t ) , W ( t ) ) , will help extend theoretical understandings and technical applications for nonequilibrium irreversible phenomena.

The time-dependent internal energy E ( t ) is assumed to be given by the liquid in a glass tube and the mechanical system coupled self-consistently to the time-dependent Equations (2.6) and (2.7), and thermodynamic work W ( t ) . The thermodynamic work should be a spontaneously extracted or externally measured quantity driven by heat, energy and entropy flows, restricted by environmental and mechanical states (temperature, friction, lubrication and wear, …) and viable energy conversion technologies. The internal energy, thermodynamic work and entropy are not given by pressure, temperature and volume in gas phase, which may be interconnected with gas-liquid and liquid-phase many-body interactions. Hence, the current nonequilibrium irreversible method is not a simple alternative expression to a gas-phase method.

The entropy is discussed as the deficit function to make calculations consistent with laws of equilibrium thermodynamics [

d E ( t ) = T ( t ) d S ( t ) + d W ( t ) . (3.1)

The time-dependent relation between the internal energy, the thermodynamic work and the equation of motion should be consistently determined. One should note that the time-symmetry in mechanics is broken; the thermodynamic work is an externally measurable quantity, not derivable in the concept of potential energy. The time-evolution of total entropy, T ( t ) d S ( t ) , is the source of time-dependent change, but the functional form is not necessarily to be known, which is determined to satisfy (3.1) at the end; this is the physical meaning of entropy as the expenditure function.

Reversibility at thermodynamic equilibrium means the work done by an engine must operate infinitely slowly, which is neither testable nor practical. The motion of up-going liquid is interpreted as the time-dependent thermodynamic work, considered to be driven by T ( t ) d S ( t ) , but because of indeterminable dissipation mechanisms, T ( t ) d S ( t ) is assumed to be derived from thermodynamic work and internal energy (3.1). The time-dependent thermodynamic work, d W ( t ) , is given by up-going liquid as,

d W ( t ) = ρ σ g z ( t ) cos θ ( t ) d z ( t ) , (3.2)

where ρ is the density of liquid (g/cm^{3}), σ is the area of a glass tube cross-section (cm^{2}), g = 980 cm / s 2 , and z ( t ) is the liquid-length in a glass tube measured from the axis of rotation; d z ( t ) is the infinitesimal change, and so, the velocity of up-going liquid is given by v z ( t ) = d z ( t ) / d t .

The constitutive equations of DB’s irreversible motion are the coupled nonlinear equations of (2.6), (2.7), and (3.1), (3.2) to determine, θ ( t ) and v z ( t ) . However, since the entropy flow, T d S / d t , is given by the internal energy and work, the equation is not complete to theoretically determine the velocity of up-going liquid, v z ( t ) = d z ( t ) / d t . From DB’s experimental observation, one can assume the constant velocity v z ( t ) = v 0 as an admissible solution, and the constant-velocity solution, v 0 , reproduced the DB motion very well [

The numerical value of the internal energy corresponding to oscillations in

The thermodynamic work flow is produced in the NIS ( t c ≤ t ≤ t d ). The energy and entropy flows, d E ( t ) / d t and T ( t ) d S ( t ) / d t , are almost similar to each other, because the amount of power flow, d W ( t ) / d t , is small compared to the one of internal energy flow.

The “thermodynamic consistency” is now ready to be shown and a NIS is measured numerically. We can calculate the time-dependent thermodynamic quantity:

τ ( t ) ≡ T ( t ) d S ( t , E ( t ) , W ( t ) ) / d t d E ( t , W ( t ) ) / d t , (3.3)

changing from a thermodynamic equilibrium to a NIS and vice versa; Equation (3.3) is also written as τ ( t ) = T ( t ) ( d S ( t ) / d t ) / ( d E ( t ) / d t ) , for simplicity.

The fundamental thermodynamic relation (1.1) is expressed by τ ( 1 ) = 1 , and the value “1” is the condition of an exact thermodynamic equilibrium state, which suggests that the entropy flow is completely converted to the internal energy flow: T ( t ) d S ( t ) / d t = d E ( t ) / d t , and so, no time-dependent thermodynamic work exists, d W ( t ) / d t = 0 from (3.1), which defines a thermodynamic equilibrium. If the value of τ ( t ) is “0”, it represents T ( t ) d S ( t ) / d t = 0 , so that a thermodynamic equilibrium produces work with no transfer or dissipation of heat, resulting in the contradiction of heat and work. The consequence is compatible with the principles of R. Clausius, W. Thomson, and the other statements on thermodynamic heat and work.

Let us denote T ˜ ( t ) , S ˜ ( t ) , E ˜ ( t ) for a NIS and T , S , E for a thermodynamic equilibrium and consider the ratio of entropy flow against energy flow: ( d S ˜ ( t ) / d t ) / ( d E ˜ ( t ) / d t ) ≡ α ˜ ( t ) and ( d S / d t ) / ( d E / d t ) ≡ α . Then, from (3.3), we have:

α ˜ ( t ) − α = τ ( t ) T ˜ ( t ) − 1 T = T τ ( t ) − T ˜ ( t ) T T ˜ ( t ) . (3.4)

It shows that α ˜ ( t ) − α > 0 , or equivalently, α ˜ ( t ) / α > 1 results in T ˜ ( t ) < T τ ( t ) . We interpret it as the amount of heat-entropy “flow-out”, compared to the thermodynamic equilibrium quantity α . Since the heat-entropy flows out of the system faster than the internal energy flow d E ˜ ( t ) / d t , the temperature T ˜ ( t ) in the NIS becomes lower than T τ ( t ) . Similarly, α ˜ ( t ) − α < 0 or equivalently, α ˜ ( t ) / α < 1 (the amount of heat-entropy “flow-in”) results in T ˜ ( t ) > T τ ( t ) . In this case, because the heat-entropy flows in the system, the temperature T ˜ ( t ) in the NIS becomes higher than T τ ( t ) .

When T ˜ ( t ) is lower than T, the nonequilibrium temperature T ˜ ( t ) will gradually increase and progress to T, ( T ˜ ( t ) ↗ T ), with T ˜ ( t ) / T < τ ( t ) → 1 , and similarly, when T ˜ ( t ) is higher than T, the temperature will gradually decrease and progress to T ( T ˜ ( t ) ↘ T ) with T ˜ ( t ) / T > τ ( t ) → 1 . The measure τ ( t ) for a NIS is consistent with observed experimental changes of T ˜ ( t ) and T. The time-dependent temperature, τ ( t ) , is constructed by the ratio of entropy transfer against the energy transfer, and it explains one of the reasons why the concept of temperature is fundamental in physics, as well as engineering and sciences in general. We will define τ ( t ) as the nonequilibrium temperature, or a measure of a NIS, which is one of the fundamental results of the thermomechanical dynamics (TMD).

Now, the second law of thermodynamics, the concept of entropy, can be explained by the statement: the nonequilibrium temperature τ ( t ) is a positive definite quantity: τ ( t ) > 0 . This is explained such that if a system is not in an equilibrium state, the entropy-flow and the energy-flow will emerge and progress to an equilibrium state. The nonequilibrium temperature converges to an equilibrium temperature ( τ ( t ) → 1 ): T ˜ ( t ) → T , which will distinguish two cases: τ ( t ) < 1 means that T ˜ ( t ) is lower than T, and τ ( t ) > 1 means that T ˜ ( t ) is higher than T. The result is physically transparent meaning that if the temperature T ˜ ( t ) in a local nonequilibrium system is initially lower than the equilibrium temperature T, the entropy will increase to the maximum entropy at equilibrium, but if the temperature T ˜ ( t ) in a local nonequilibrium system is initially higher than the equilibrium temperature T, the entropy will decrease to the minimum entropy at equilibrium. One should note that the revolution to extrema of entropy depends on the difference between T ˜ ( t ) of a NIS and the corresponding equilibrium temperature T.

The internal energy, E ( t ) , and the energy-flow, d E ( t ) / d t , of the 1st and the 2nd dipping motions are shown in

The DB’s state is in mechanical and thermodynamic equilibriums at the outset and then, the transition to a NIS arises at t c ≃ 24.0 ( s ) . The straight horizontal line represents a thermodynamic equilibrium, and the behaviors at t c and t d are only discontinuous in our model simulations because of the use of piecewise continuous step functions for the simplification of the physical mechanism at t c and t d ; for instance, one can integrate the heat-conduction mechanism at t c and t d , in order to elaborate the conduction of time-dependent temperature. The nonequilibrium temperature, τ ( t ) , changes rather like a heat-wave pulse function, and one should note that τ ( t ) is not necessary a monotonically changing function, except at a near-equilibrium. After the DB’s 1st-drinking at t 1 d ≃ 38.8 ( s ) , the state decouples to equilibrium states and returns back to oscillations around θ = 0 and T ∂ S ( E , V ) / ∂ E | V , N , ⋯ = 1 , ( t 1 d < t ). The second NIS arises in the time period, 60.5 < t < 68.3 ( s ) , and the nonequilibrium temperature, τ ( t ) , shows oscillating evolutions to an equilibrium temperature, as shown in

The thermal conduction mechanism could be integrated in the current thermomechanical method. There would be so many applications, for example, the irreversible processes such as friction, ignition, combustion and detonation mechanisms [

The method of thermomechanical dynamics by employing a drinking bird mechanism has shown useful concepts and specific results for nonequilibrium irreversible states, which is different from probability theory and distribution function method in kinetic theories. The results and the foundations of TMD are summarized in this section.

The self-consistent analysis of the DB system would simultaneously direct one to confront emergent phenomena: the coupling or decoupling between mechanics and thermodynamics. It is never a simple, complementary mechanism explained by mechanics and thermodynamics. It demands the transition from a time-symmetric state to a time-symmetry-broken state and vice versa. Both mechanics and thermodynamics will break down, and a new state emerges as a NIS. The time-symmetry of Hamiltonian is broken, resulting in non-conservation of energy, but instead, the total heat and energy is conserved in the time range, t c < t < t d . Mathematically speaking, the equation of motion for DB’s mechanical equilibrium is given by a nonlinear differential equation with constant coefficients, but the equation of motion in the corresponding NIS is given by the same nonlinear differential equation with time-dependent coefficients. The time-dependent nonlinear equation has independent solutions not derivable from the one with constant coefficients. The transformation to a nonlinear differential equation with time-dependent coefficients might correspond to a mathematical way of expressing a transition from a physical phase to a new emergent phenomenon. The nonlinear differential equations with constant and time-dependent coefficients must be mathematically categorized in a different class of nonlinear equations. This is an important mathematical property discovered in the analysis of the DB system.

The concept of the local equilibrium and the near-equilibrium state, the local microscopic reversibility, the linearity of fluxes and forces in transport processes [

Based on discussions and results so far, the methodology and the fundamental postulates of TMD for heat engines and transport mechanism will be summarized as follows:

1) If thermal and mechanical equilibriums coexist like a system of heat engine, time-dependent Hamiltonian (or Lagrangian) and equations of dissipative motion (with friction and friction-induced time-dependent phenomena) must be constructed, which should be identified as the time-dependent internal energy and the corresponding nonlinear equation of dissipating motion.

2) The work, d W ( t ) , must be an externally measurable quantity and defined against the dissipative mechanical motion proceeding to mechanical equilibrium, given by the inner product: F ( t ) ⋅ d l ( t ) ( d l ( t ) ; the work-direction increment), or, by available energy transformation technologies such as heat-electricity, charge-energy transfer and so forth.

a) The time-dependent force F ( t ) is local, not given by a mechanical potential defined in a given Hamiltonian. The time-symmetry of mechanical Hamiltonian is broken by the generation of time-dependent work, d W ( t ) .

b) The time-dependent Hamiltonian H ( t ) identified as the internal energy E ( t ) , and the time-dependent thermodynamic work d W ( t ) , must maintain the total heat-energy conservation law: d E ( t ) = T ( t ) d S ( t ) + d W ( t ) , and the differential form of the time-dependent Gibbs relation in general. The total entropy T ( t ) d S ( t ) is interpreted as the expenditure function to make TMD consistent with the total heat-energy conservation law.

3) The measure of a NIS, or the nonequilibrium temperature is defined by:

τ ( t ) = T ( t ) d S ( t ) / d t d E ( t ) / d t . (4.1)

a) Thermodynamic equilibrium, the relation between internal energy and work when d S ( t ) / d t = 0 , and the principle of entropy correspond to τ ( t ) = 1 , τ ( t ) = 0 , and τ ( t ) > 0 , respectively.

b) The thermal and mechanical equilibriums can exist when the entropy flow could not affect the internal energy to produce observable work. Near equilibriums and not-far-from equilibriums are studied by τ ( t ) = T ( t ) ( d S / d t ) / ( d E / d t ) ∼ 1 .

The postulate discussed above is applicable to examine heat engines and NISs, and it should be extended by including heat-conduction, ignition and detonation, friction and diffusion mechanism for elaboration and applied to self-lubricating, self-organizing smart materials and so forth [

We have shown the theory of nonequilibrium irreversible thermodynamics and the mechanism of a drinking bird as a rigorous and fascinating device and obtained new mathematical and physical results: the coupling and decoupling of thermal and mechanical states, symmetry-breaking and restoration, bifurcation solutions and independent solutions of nonlinear equations with time-dependent coefficients, the measure of nonequilibrium states and the postulate of thermomechanical dynamics of nonequilibrium irreversible phenomena.

A DB toy is composed of a many-body system of interactions among liquid, gas and interface of gas-liquid. The TMD demands that thermomechanical variables should be carefully and optimally chosen to be minimal because constituent variables will be self-consistently intertwined with others. Hence, one should be careful that the partition to many constituents and the direct sum of constituents usually discussed at thermodynamic equilibrium are not allowed in NISs. The measure of irreversibility demands that the time-dependent fluctuation of entropy should be evaluated relatively to the time-dependent internal energy. It could be physically reasonable that the measurement-induced fluctuations, ignition and detonation, friction and diffusion mechanism, extremely high-temperature gradients at small scales and short periods of time should be measured and studied by the ratio of entropy and internal energy flows. Heat conduction and thermal management in modern nanostructure technologies at molecular and atomic levels may require new kinds of experiments and thermomechanical methods for measurement and analysis [

In the paper [

The mechanism and fundamental characteristics of a drinking bird are the scientific motivation for our writing a series of papers, but in addition, it directed us to scientific ways of thinking, supported and encouraged us while studying a physical model of heat engines. The scientific views that should be fundamental to discuss here are: reductionism, structuralism and emergentism (or emergent dynamics). The intuitive understandings of the three views are essential for scientists as a whole. The successful reductionist views but associated contradictions and conceptual problems induced in diverse fields of science, nature and human society, are well summarized in the papers [

The wholeness is an abstract concept difficult to define and often seriously misunderstood in sociology and politics. For simplicity, let us consider the equation of motion and Hamiltonian of a simple pendulum shown in (2.4) and assume that the basic building blocks for the pendulum Hamiltonian are length and mass (or moment of inertia), gravity and angle, ( l , m , g , θ ). The wholeness appears as symmetries and the energy conservation law for basic building blocks. Then, one would introduce the coupling of many pendulums with possible potential interactions to explain dynamics of the system. However, it is generally known that many-body interaction phenomena such as superconductivity, collective phenomena, thermodynamic consistency [

As the reductionist and structuralist views are considered to be synthesized in physics as the rigorous methodology of Lagrangian and Hamiltonian mechanics, the view against reductionism in physics has gradually surfaced from microscopic many-body interactions [

The examples against reductionist views are abundant in condensed matter physics, such as superconductivity, phase-transition phenomena, symmetry-breaking, effective field theories in hadron dynamics and so forth. The assertion coincides with one of the DB’s claims for nonequilibrium irreversible states: d S ≠ ∑ i d S i , with the indication that the relation, d S = ∑ i d S i , is only true at thermodynamic equilibrium and a quantum statistical ensemble when interactions are negligible among constituents. The logical character supports the current method of TMD for the nonequilibrium irreversible dynamics. The reductionist way of thinking may conclude that even emergent phenomena and structuralism will be resolved from a theory of everything, which comes from typical misunderstandings of emergent phenomena and structuralism. Although the reductionist method has contributed to science and technologies, “The existence of emergent phenomena undermines the kind of reductionism that is presupposed in the search for a theory of everything” [

The technologies and scientific ways of thinking have contributed to human societies and understandings of nature from microscopic to macroscopic phenomena, liberating our mind and spirit from oppressions and limitations. The achievement of sciences and technologies is powerful and gigantic, but a big science with sociology, politics and group-oriented authoritative ways of thinking may suppress our scientific mind and spirit [

The authors declare no conflicts of interest regarding the publication of this paper.

Uechi, H., Uechi, L. and Uechi, S.T. (2021) Thermodynamic Consistency and Thermomechanical Dynamics (TMD) for Nonequilibrium Irreversible Mechanism of Heat Engines. Journal of Applied Mathematics and Physics, 9, 1364-1390. https://doi.org/10.4236/jamp.2021.96093