Dissipative Dynamics I

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Accommodate the Thermodynamic Arrow of Time1. H.R. Crecraft – [email protected]. Abstract. To understand the collapse of the wave function, ...
Dissipative Dynamics I: Generalizing Physics’ Conceptual Model to Accommodate the Thermodynamic Arrow of Time1 H.R. Crecraft – [email protected]

Abstract To understand the collapse of the wave function, the thermodynamic arrow of time, and the evolution of complexity, we need to integrate physics and thermodynamics into a single integrated conceptual framework. This essay describes the collapse of the wave function and the thermodynamic arrow of time within the non-equilibrium thermodynamic conceptual framework. The universe has evolved to its current exceptional state, where we are here to ponder why. Physics tells us that the universe must have started in an exceptionally improbable initial state, or perhaps there are multiple universes but we perceive only this exceptional one. These explanations are a consequence of the prevailing interpretations of modern physics, which assume that nature is fundamentally reversible. Reversibility means there is no spontaneous selection of possibilities, and the future, as well as the past, is determined and set in stone. Observations, however, are intrinsically indeterminate and statistical. We can observe the spontaneous decay of a high-energy particle to its more stable low-energy state. Prior to observation, the quantum wave function describes the isolated particle as a superposed state, comprising a simultaneous superposition of its high-energy and low-energy states. At observation, we invariably observe only one state or the other. This is the collapse of the wave function to one of its constituent and definite states. This is commonly referred to as the measurement problem. The measurement problem underlies the conceptual difficulties of quantum mechanics that have existed since its beginnings2. Physics asserts that the collapse of the wave function is triggered by the measurement process or some other external perturbation. It asserts that, as long as a system remains isolated from external fluctuations, there can be no indeterminate irreversible change. Since the universe (or multiverse), by definition, has no surroundings, it follows that it must be fundamentally reversible and its future fully determined. Furthermore, physics attributes the apparent randomness of external fluctuations to incomplete information on the system’s surroundings and its interactions. Physics thereby ascribes indeterminism and the appearance of randomness to our incomplete knowledge. Physics treats time simply as a coordinate along the time axis of four-dimensional spacetime. Events unfold the way scenes unfold in a movie. We can progress through a movie film or along the time axis of our universe. We can observe events as we proceed, but the film itself, and the universe in its four-dimensional space-time, are both static. In such a universe, there is no spontaneous selection of possibilities, no evolution, and no free will. Everything that will be, already is. Reversibility is not an inevitable consequence of the laws of physics, however. Newton’s laws of classical mechanics do not distinguish between past and future and they do not define an 1

arrow of time. They nevertheless accommodate irreversibility. Newtonian mechanics is all about forces and motion, and friction is simply a force that dissipates the kinetic energy of motion. Laws of physics are ultimately based on observations. We observe that real pendulums do not swing forever. They irreversibly slow down and stop. We can watch a drop of ink irreversibly disperse into a glass of water. Quantum observations are intrinsically indeterminate and statistical. When we observe and measure an unstable quantum particle, it is impossible to predict the outcome, even if we completely know the particle’s initial state. All that we can do is to accurately predict the statistical probabilities of quantum measurements. Yet physics asserts that a system, isolated from external perturbations, evolves reversibly and deterministically. The nature of a system as it exists in isolation, however, cannot possibly be resolved by observation or measurement. Observation and measurement necessarily involve an external interaction with the system, and this violates the condition of isolation. The nature of a system as it exists behind the veil of isolation is strictly a matter of interpretation of experimental results. So why does modern physics assume reversibility and determinism as fundamental and intrinsic attributes of nature? In 1834, a hundred and fifty years after Newton, William Rowan Hamilton reformulated Newtonian classical mechanics3. Hamilton assumed that a system is resolvable into a collection of elementary particles. An elementary particle can have kinetic and potential energies, but it has no internal parts and so it has no internal energy (distinct from its potential energy). A system’s energy is equal to the sum of the energies of its parts. The system’s energy therefore consists only of potential energy and kinetic energy. Potential and kinetic are available in principle for useful work. By the conservation of energy and the equivalence between energy and work potential, work potential is therefore conserved. This is the conceptual model of reality that underlies Hamiltonian classical mechanics, quantum mechanics, and relativity. The Hamiltonian conceptual model marked a fundamental break from the mechanics of Newton, which could accommodate friction, dissipation, and loss of work potential. Nowhere in Newton’s formulation of classical mechanics is there the conservation of work potential. Hamilton’s conceptual model of physical reality set physics down a path that regards nature as fundamentally reversible and deterministic. The Hamiltonian conceptual model rejects thermal randomness and indeterminism as fundamental properties. Eliminating dissipation and conserving work potential means that the mechanical state of a system and its surroundings at any instant fully determines the system for all time. The future, as well as the past, is set in stone. Numerous experimental results have fully validated the theories of modern physics. The experimental support of quantum mechanics, relativity, and statistical mechanics is not questioned. The Hamiltonian conceptual model, which provides the basis for interpreting experimental observations, is also widely accepted and rarely questioned. Experimental results are facts that we can all agree on, but the Hamiltonian conceptual framework is an interpretation of statistical experimental results that presumes nature as fundamentally reversible and deterministic. The Hamiltonian model interprets the indeterminism of quantum observations and the collapse of the wave function as due to external perturbations. External perturbations only appear to trigger random change because we do not know their details. Similarly, the flow of heat or the diffusion of ink into water only appears to be irreversible, because we lack knowledge of the 2

microscopic details of the system’s fundamental particles and their interactions. Because of these uncertainties, a system’s path cannot be precisely determined. We can only describe it statistically. Statistical mechanics expresses the uncertainty in the microscopic details as the Boltzmann entropy, and it equates this to the equilibrium thermodynamic entropy. This describes Edwin Thompson Jaynes’s mechanical views of entropy and statistical mechanics—that they are not about the system, but rather about our knowledge of the system4. Thermal randomness is regarded as just an artifact of our imperfection of measurement and observation, and the result of unknown, but deterministic, interactions. The Hamiltonian conceptual model is consistent with experimental facts and with the existing theories of modern physics and statistical mechanics. Mere compatibility with observations, however, does not validate the model’s interpretation of experimental facts. A conceptual model does not make predictions. A model merely provides a simplified representation of reality. It provides the conceptual framework that we use to interpret the experimental results to verify or refute a theory. Other interpretations of the same observations may be equally compatible with experimental results. The Hamiltonian model presumes reversibility and determinism at the level of a system’s fundamental particles and forces. Reversibility and determinism lead to the conceptual difficulties with the quantum measurement problem and the indeterminate collapse of the wave function. It leads to a universe whose entire history is determined at the instant of its creation. We seek a better way. We reject a future devoid of possibilities and choices. We reject the Hamiltonian conceptual model. In place of the Hamiltonian conceptual model, we introduce the first three postulates of dissipative dynamics: Postulate 1: Absolute zero does not exist. No system has surroundings at absolute zero and no system can be perfectly isolated from its ambient surroundings. Postulate 2: If a system is in thermal equilibrium with its surroundings, then a complete description of the system’s equilibrium state is defined by properties measurable by a perfect observer in the ambient surroundings. Postulate 3: The state of a non-equilibrium system is completely defined by a reference state in equilibrium with an ambient observer and by perfect measurement. Perfect measurement is defined as a reversible and deterministic process of transformation between the system’s state and the reference state. These postulates recognize that absolute zero does not exist. Even the universe itself has ambient surroundings defined by its cosmic background radiation at 2.7 kelvins. A positive ambient temperature generates thermal randomness within the surroundings and within the system itself. These postulates accommodate a model that we designate the non-equilibrium thermodynamic model. This model describes a system as it exists with respect to equilibrium surroundings at a fixed ambient temperature. The non-equilibrium thermodynamic model is a generalization of both the Hamiltonian and the equilibrium thermodynamic conceptual models. The Hamiltonian model is the idealized special case in which the ambient temperature is assumed to be absolute zero. This idealization can be approached, but it can never be attained. The Hamiltonian conceptual model eliminates random 3

thermal fluctuations and the source for indeterminate change. Given this (non-attainable) idealization, Maxwell’s Demon5 is free to reversibly extract work from the kinetic energy of a gas’s particles. All process is reversible and deterministic. Equilibrium thermodynamics is another idealized special case. In equilibrium thermodynamics, the system has a uniform and fixed temperature equal to the temperature of its surroundings. In equilibrium thermodynamics, free energy defines the system’s potential work on a reference at the system temperature. Free energy is defined independent of the system’s surroundings and it is a property of state within the equilibrium thermodynamic conceptual model. The non-equilibrium thermodynamic model also goes beyond the conceptual model for classical irreversible thermodynamics (CIT). CIT defines the free energy of a non-equilibrium system simply as the sum of free energies of the parts. For a non-isothermal system, however, the temperature varies over space and time. The free energies of the system’s parts describe work potentials as defined with respect to differing and changing local reference temperatures. The nonequilibrium thermodynamic model, in contrast, defines a system’s work potential by its exergy. Exergy is the potential work on a reference in the system’s actual ambient surroundings. Exergy, but not free energy, can describe a non-equilibrium system as a single integrated system with respect to a single ambient reference state. Exergy, but not free energy, constitutes a property of state within the non-equilibrium thermodynamic conceptual framework. In the non-equilibrium thermodynamic model, a system’s state cannot be defined in isolation. As described in dissipative dynamics’ postulates, a state can only be defined with respect to the system’s external ambient reference. We can view this as an extension of Einstein’s special theory of relativity, which showed that physical reality can only be defined relative to an external inertial reference. Dissipative dynamics simply adds an ambient temperature to this inertial reference. Within the non-equilibrium thermodynamic model, the Second Law of thermodynamics asserts that exergy is irreversibly dissipated into ambient heat, and the First Law states that the total energy (exergy plus ambient heat) is conserved. Ambient heat (Q) is the thermal energy at ambient temperature, and it is also a property of state. Ambient heat has no capacity for work on the ambient surroundings, and it exists as a third form of energy, distinct from the potential and kinetic energies of the Hamiltonian model. It is closely related to entropy (S) by Q=TaS, where Ta is the ambient temperature of the surroundings. We can see from this simple relationship that, for a fixed ambient temperature, the irreversible dissipation of exergy to ambient heat and the irreversible increase in entropy are equivalent interpretations of the Second Law. The interaction of a non-equilibrium system with its uniform and constant environment irreversibly leads the system toward equilibrium with its surroundings, by spontaneous and indeterminate transitions from states of higher exergy to states of lower exergy. This defines the irreversible dissipation of exergy and the thermodynamic arrow of time. In the thermodynamic limit, dissipation is continuous. This describes, for example, the irreversible dispersion of a chemical or thermal gradient. In quantum mechanics, in contrast, exergy transitions are discrete. This occurs, for 4

example, during radioactive decay or the transition of an electron to its ground state, where the emitted energy is dissipated and lost to the ambient surroundings. Within the framework of dissipative dynamics, thermodynamics and quantum mechanics both describe the dissipation of exergy to heat at the system’s positive temperature. In thermodynamics, dissipation and the passage of irreversible time are continuous. In quantum mechanics, dissipation is associated with the irreversible collapse of the wave function, and it is discrete and quantized. The reversibility of time along the time axis of space-time is segmented into reversible intervals, separated by irreversible transitions. The Hamiltonian conceptual model applies to the reversible intervals, but it fails to address the discontinuities. The quantum measurement problem arises because the Hamiltonian conceptual framework assumes the surroundings to be absolute zero and it cannot accommodate the dissipation of energy during a quantum transition. Instead, it describes irreversibility and indeterminism as phenomenological, associated with imperfect measurement or its perturbative effects, and not as properties of the system as it actually exists in isolation. The non-equilibrium thermodynamic model embraces heat, entropy, and dissipation as intrinsic properties of state for the system and not merely as phenomenological properties related to an observer’s knowledge or the measurement process. The non-equilibrium thermodynamic model is a generalization of the Hamiltonian model, and it more closely represents a physical system, as it exists with respect to its actual surroundings at a positive ambient temperature. The non-equilibrium thermodynamic model is still essentially a model of states. It can describe the exergy of non-equilibrium states. The second law describes the relative stability of states of lower exergy over states of higher exergy. It describes the tendency of states to dissipate exergy and to transition to states of lower exergy and higher stability. It explains the thermodynamic arrow of time, but it cannot explain the stability of irreversible processes or the emergence of complexity. In a follow-up article, we will describe a further generalization, the dissipative dynamics conceptual model. This model still assumes that the surroundings are stationary, but it allows the surroundings to be non-equilibrium. Within this more generalized framework, we will be able to define the stability of irreversible process. This will allow us to define a second arrow of time, the arrow of progress, leading to the spontaneous evolution toward complexity. The dissipative dynamics model is another step forward in extending our conceptual model of reality to make it more broadly applicable to real systems as they exist in real environments. Notes and References for further Reading

This article is based on Crecraft, H. R. 2017. Evolving Complexity—Time’s Arrows and the Physics of Emergence, Symbion Publishing IBSN: 978-0-9986178-0-0. See www.evolvingcomplexity.com for extended excerpts. 2 For a review of quantum mechanical interpretations, see https://arxiv.org/abs/1408.2093. 3 W.R. Hamilton, "On a General Method in Dynamics.", Philosophical Transaction of the Royal Society Part II (1834) pp. 247–308; Part I (1835) pp. 95–144. 4 http://charlottewerndl.net/Entropy_Guide.pdf. 5 see https://en.wikipedia.org/wiki/Maxwell%27s_demon for discussion and references 1

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