Entre el tiempo y la eternidad is a work of theoretical physics and philosophy of science, written by Ilya Prigogine with Isabelle Stengers and first published in French in the late 1980s, then translated into Spanish by Javier García Sanz for Alianza Editorial. Across eight chapters and three technical appendices, it confronts a single stubborn question: why did physics, alone among the sciences, deny the reality of time, and how can it now take time seriously at the most fundamental level? The book is not a survey of modern physics but a sustained argument, moving from Boltzmann’s nineteenth-century failure through chaos theory, dissipative structures, quantum measurement, and cosmology.
“En todos los niveles, escribíamos, la ciencia redescubre el tiempo.”
The Core Argument
The book’s central claim is that the classical tradition in physics—from Galileo and Leibniz through Newton, Lagrange, and Hamilton—built its ideal of intelligibility on a negation of time. The principle of sufficient reason, in Leibniz’s formulation, equates a “full” cause with a “total” effect, and that equivalence implies reversibility: a body that loses height on an inclined plane gains exactly the velocity needed to climb back. A deterministic, reversible trajectory permits no intrinsic distinction between past and future. This was not a minor technical assumption but the founding principle of dynamics, and it made the arrow of time, so evident in experience, appear illusory.
Boltzmann tried to give irreversibility a dynamical meaning. His H-theorem of 1872 modeled a gas as a population of colliding particles whose collisions progressively erase any initial difference from equilibrium, producing a monotonic approach to uniformity. The analogy with Darwin is deliberate: both replaced individuals with populations and linked small variations to long-timescale collective change. But where Darwin explained the emergence of increasingly complex adapted species, Boltzmann described evolution toward sameness—and his interpretation collapsed under criticism. Loschmidt’s velocity-inversion objection showed that for every entropy-increasing evolution, the reversible laws permit an inverse evolution restoring what was destroyed. Boltzmann was forced to concede that no privilege could be attributed to entropy-increasing processes, and he retreated to a probabilistic interpretation in which irreversibility reflects only our ignorance of microscopic detail.
Prigogine and Stengers argue that this defeat is now reversible. The discovery of unstable and chaotic dynamical systems undermines the objections that defeated Boltzmann. For sufficiently unstable systems, a time horizon exists—set by the Lyapounov time—beyond which no determinate trajectory can be attributed to the system. Poincaré’s recurrence theorem and Loschmidt’s inversion both require times far exceeding this horizon, so they lose their pertinence. The authors’ path is deliberately inverse to Bergson’s: rather than starting from intimate experience of duration, they start from evidence immanent to physics itself—phenomenological descriptions affirm the arrow of time, and fundamental theory should be questioned in light of that evidence, not the reverse.
The argument extends beyond classical dynamics. In quantum mechanics, the formalism’s dual structure—reversible evolution of the wave function coupled to irreversible measurement—has generated paradoxes from Schrödinger’s cat to the quantum Zeno effect. The authors contend that irreversibility is not introduced arbitrarily by observation but is a necessary condition of knowledge, and that the atom’s apparent stability reflects the small value of the fine-structure constant in our “warm” Universe. In cosmology, the Big Bang has made the Universe a genuine object of physical science, and the primordial Universe points toward a physics of intrinsically irreversible processes. The book’s conclusion is that the arrow of time is universal, articulating descriptions of the Universe at all levels, and that the old opposition between “phenomenological” and “fundamental” descriptions must give way to a new coherence.
Inside the Book
The Genesis of the Book
The Spanish edition of Entre el tiempo y la eternidad, translated by Javier García Sanz and published by Alianza Editorial, gathers an introduction, eight chapters running from “El tiempo cuestionado” to the title essay, three appendices, and an analytical index. The authors explain that the project began as a preface to a collection of articles by one of them, describing how the perspectives opened by La nueva alianza had developed. It grew instead into the book whose necessity they had already announced in that earlier work’s pocket-edition prologue, where they wrote that science rediscovers time at every level and that another book should sketch this renaissance. Some theoretical problems central to their approach have since moved quickly, and they prefer to describe these developments “live,” even before publication in specialized journals. This is therefore not a continuation of La nueva alianza but a more radical restatement of the problem of time.
La nueva alianza had renewed the phenomenology of time through the thermodynamics of systems far from equilibrium: the inversion of the classical paradigm that identified entropy growth with disorder, the constructive role of irreversible phenomena, self-organization far from equilibrium, and the “dissipative structures” that help us understand life. It carried the debate into classical dynamics as well, showing that the renaissance of dynamics tied to unstable dynamical systems challenges the determinist ideal, leading from determinism to probabilities and from reversibility to irreversibility—conclusions once thought revolutionary, now increasingly accepted by physicists.
The question of time now presses urgently in quantum mechanics and cosmology, the twentieth century’s leading sciences. Despite their revolutionary character, both remain direct heirs of classical dynamics regarding time, enclosing a radical negation of irreversible time. The authors propose modifying these conceptual structures. They recall Heisenberg’s distinction between the abstract painter, maximally original, and the theoretical physicist, maximally faithful to tradition and modifying it only when forced—a situation they claim to inhabit now, driven by internal constraints of physics rather than by conviction. They acknowledge the boldness of questioning the two most fertile sciences, but insist there is no turning back: the illegitimate ideal of intelligibility has been surpassed. The twentieth century compares only to the seventeenth; it began with relativity and quantum mechanics, yet these stay within the classical tradition—a continuity challenged by unexpected experimental discoveries: elementary particles, non-equilibrium structures, and the evolution of the Universe, all pointing toward overcoming the negation of irreversible time. A new coherence becomes visible: at every level of physics we find the time the classical tradition denied.
The book’s scope is broader than La nueva alianza; its guiding thread is the singularity of physics—its diversity of theoretical approaches and their deep solidarity—not its history. It is not an introduction to contemporary physics: no general exposition of dynamics, thermodynamics, quantum mechanics, or relativity, nor of symmetry and the unification of fundamental interactions; for dissipative systems they draw mainly on problems studied by their own group. Three appendices require some technical knowledge. The authors hope readers will discover how foreign physical theories are to the “almost imperialist” image often held of them, revealing their character as bold, risky invention and creation of meaning—explaining how physics could deny time while phenomenological descriptions affirm it, and how it can now include it at the most fundamental level.
Boltzmann’s Drama
La nueva alianza used the history of physics as its thread; Entre el tiempo y la eternidad explores the confrontation between physics’ conceptual schemes and the problem of time, starting from Boltzmann’s work—the first act of this confrontation, when physicists understood the radical constraints classical dynamics imposed on thinking becoming. For most physicists after Boltzmann, the truth of physical time must be defined at the level of fundamental theories, from which the status of time in phenomenological descriptions derives. The authors’ path is inverse, in a sense parallel to Bergson’s, for whom the starting point of all thought seeking reality must be our most intimate experience—”the feeling we have of our own evolution and of the evolution of all things in pure duration.” Yet their starting point is not intimate experience but an evidence immanent to physics itself: not only do phenomenological physical descriptions affirm the arrow of time, they now lead us to understand a world in becoming, where the emergence of the new acquires irreducible significance. Instead of deducing phenomenological time from “fundamental” time, they question the conception of physical time in fundamental theories starting from phenomenological evidence.
This inversion raises the question of the authority historically conferred on these theories: how could a merely technical problem—the impossibility of defining a function that a dynamical system’s evolution makes grow with time, as with entropy—give rise to the almost inconceivable claim that the difference between past and future has no objective reality? How to understand the cultural singularity that allowed physics to deny such overwhelming evidence, which no culture had questioned before? The first two chapters address this. Chapters 3 and 4 explore the contemporary development of the science of dissipative, entropy-producing processes, which after Boltzmann was interpreted as relative only to the approximate character of our observations and calculations. They show how this science has become a science of becoming—not by providing a general model of becoming for other sciences, but by discovering, from its own objects, conditions without which no evidence, biological or social, can define becoming: irreversibility, probabilities, coherence.
Boltzmann called the nineteenth century “the century of Darwin,” who showed the invariance of living species to be mere appearance. Yet for most physicists Boltzmann’s name evokes an antithetical achievement: showing irreversibility to be appearance, beyond which reigns the invariant eternity of dynamical laws. Boltzmann’s drama: he attempted in physics what Darwin did in biology—giving meaning to evolutionary time at the fundamental level—but reached a dead end, capitulating to dynamics’ demand for eternity. Darwin transformed biology’s object, showing that studying living populations and their history, not individuals, reveals how individual variability under selection generates species drift. Similarly, Boltzmann sought to show that in a numerous particle population, collisions give meaning to entropy growth and thermodynamic irreversibility.
In 1872 Boltzmann presented his H-theorem. The H function translates collision effects; constructed to decrease monotonically to a minimum, it reaches a velocity and position distribution no longer modified by later collisions—a microscopic model of irreversible evolution toward equilibrium. Collisions progressively erase any initial difference from the equilibrium distribution. Both theories replace individuals with populations, linking small variations to long-timescale collective evolution—but Darwin explains increasingly complex adapted species, while Boltzmann describes evolution toward uniformity. Darwinism triumphed; Boltzmann’s interpretation succumbed to criticism. Boltzmann was cornered into admitting he had failed to give a microscopic interpretation of irreversibility. An antithermodynamic evolution, with entropy decreasing and differences deepening, remained physically possible.
Boltzmann was convinced that understanding the Universe means understanding its historical character; irreversibility defined by the second principle had fundamental meaning. But he was heir to dynamics’ tradition, which conflicted with any intrinsic arrow of time. Retrospectively, his failure seems evident: a trajectory is deterministic and intrinsically reversible, permitting no past and future distinction; explaining the irreversible by the reversible is, as Poincaré noted, an error logic alone condemns. Today we know how to overcome this contradiction, but only via a fundamental transformation of dynamics’ conceptualization.
The Principle of Sufficient Reason
The passage asks how Boltzmann and his contemporaries hoped to give dynamic meaning to irreversibility, and why they did not see that dynamical time affirms both deterministic causation and the essential equivalence of time’s two directions. The idea of dynamical reversibility is almost inconceivable: time’s question has always centered human concerns.
The Horizon of Predictability
Lyapounov time gives real meaning to the phrase “two identical systems.” Once a system evolves for a span long compared with that time, knowledge of its initial state stops determining where it goes. Chaotic systems therefore possess a temporal horizon—displaceable but never abolished. Pushing it back grows ruinously expensive: extending predictability tenfold demands precision greater by a factor of e¹⁰. Beyond the horizon, evolution can only be described as erratic behavior shared by every system shaped by the same chaotic attractor. Bernard D’Espagnat’s work on “empirical” causality and intersubjectivity cannot capture this, since chaotic behavior assumes no ceiling on measurement sophistication.
Physicists and mathematicians have mapped many chaotic systems and the routes into chaos. Take the weakly dissipative spherical pendulum whose suspension point moves periodically, studied by John Miles and expounded by James Lighthill. A first bifurcation arrives at T = 0.989 T₀, a second at T = 0.99887 T₀; further bifurcations pile up subharmonics until, at T = 1.00234 T₀, regularity vanishes entirely and the motion turns erratic, the earlier choice forgotten. The transformation is general, touching weakly dissipative systems and purely dissipative chemical ones like the Belousov-Zhabotinsky reaction.
Probabilities often merely register ignorance—births, deaths, marriages, divorces. Climate poses harder questions. Two to three hundred million years ago the climate was milder and the oceans some eighty meters higher; forty million years ago the pole-equator contrast sharpened; the Quaternary brought glaciations; seven thousand years ago a climatic optimum ended three thousand years ago with the Iron Age, when the Sahara still supported prosperous agriculture. Recognizing chaotic attractors matters for human action on the environment. Grassberger and Procaccia’s method extracts a dimension from a single variable’s time series; the Nicolises applied it to O¹⁸ isotope temperatures from equatorial Pacific sediments, suggesting a chaotic attractor of dimension 3.1, so four variables might explain Earth’s climatic history. Brain activity tells a similar story: deep sleep shows deterministic chaos with a fractal attractor of five variables, wakefulness shows none, and epileptic seizures show a fractal attractor of perhaps only two—epilepsy as excessive regularity. Economic data analysis, by contrast, remains complicated.
Irreversibility as Creator
Irreversibility itself turns creative. Snowflakes form regular spheres near equilibrium and branches far from it; crystals carry the memory of their formation path; chemistry, as Berthelot said, “creates its object.” On the origin of life, biomolecules appear as both actors and products of dissipative metabolism, possibly arising at submarine hot springs along active ridges, where water at 275 atmospheres and 350°C meets cold ocean water—intense non-equilibrium. “In the beginning was activity”: life as daughter of the Earth before daughter of the Sun.
Information theory sharpens the puzzle. Shannon’s measure captures “surprise,” maximal for random sequences—Borel’s typing monkey. But a text is not purely random. Of the 20¹⁰⁰ possible protein sequences of a hundred amino acids, how could some acquire meaning? Chaitin and Kolmogoroff measure information as program length, yet presuppose what is indispensable, so the measure is not neutral. Markov chains offer a middle path, neither random nor deterministic. Nicolis and Subba Rao modeled three monomers transforming into one another and aggregating into a polymer chain. Near equilibrium the chain is Shannon-maximal; with an oscillating chemical clock it becomes periodic XYZXYZ; with a chaotic attractor it acquires Markov properties—non-repetitiveness, long-range correlations, broken spatial symmetry. A simulation of some ten thousand units yields hypersymbols ZYX, ZXYX, ZX, excludes subsequences XX and YXY, and realizes only 21 of 2,187 possible seven-letter sequences. Irreversibility can thus inscribe itself in matter, creating genuinely new sequences with intrinsic sense—not a reduction of history to physicochemical laws but a revelation of irreducible richness. “Markovian” molecules explain only the terrain, not history. A new chapter in chemistry opens.
The Message of Entropy
Boltzmann and Bergson open the chapter on entropy’s message. Classical dynamics’ denial of time, once thought definitive, has been overturned by the discovery of chaotic systems. James Lighthill, president of the International Union of Theoretical and Applied Mechanics, offered a solemn declaration: the enthusiasm of predecessors over Newtonian mechanics bred generalizations about predictability now known to be false, and he apologized collectively for misleading the cultivated public about determinism from 1960 onward. Determinism, once ineluctable, is now valid only in particular cases; probabilities, which Boltzmann treated as an auxiliary for contingent ignorance, acquire intrinsic significance.
The exponential divergence of trajectories, the positive Lyapounov exponent, the temporal horizon—these apply even to the simple spherical pendulum. Lighthill notes that regular behavior turning chaotic is not new, but fluids and Boltzmann’s model involve huge numbers of particles, letting irreversibility be blamed on approximations. Simple, exactly treatable systems yielding chaos are therefore fundamental: there is no escape, and the physicist can decipher entropy’s message. Why does entropy exist? The coin-toss example separates probability as ignorance from probability as fundamental: with finite precision, rational and irrational points interleave, so no initial condition deterministically yields one outcome. Leibniz’s passage from finite to divine knowledge and Duhem’s “margin of physical error” return here.
The baker’s transformation—a unit square flattened, halves superposed—is reversible yet unstable, fragmenting any region so nearby points diverge. Its horizontal coordinate is dilatant, its vertical contractant, distances doubling per iteration with Lyapounov exponent log2. Unlike earlier attractors, this chaos arises under a reversible law. The objections that defeated Boltzmann—Poincaré’s recurrence theorem, the inversion thought experiment—lose pertinence, since Poincaré’s time far exceeds Lyapounov time. Smoluchowski’s statement that over immensely long observation all processes appear reversible is cited, and computer calculations with time-reversible equations confirm irreversibility for sufficiently long times.
Bernoulli Shifts and the New Language
The baker’s transformation works through Bernoulli shifts: binary decimals of coordinates shift index positions, each decimal independent and equiprobable like coin tosses. Periodic trajectories correspond to rational points and are the exception; ergodic trajectories cover phase space. Physicists can never specify a point, only a finite-decimal region—a window of finite n. Each transformation pushes unknown decimals toward u₀; either precision shrinks or probabilities enter, until after 2n+1 transformations the system is anywhere with equal probability. Meteorology resembles this: insignificant details escaping observation may later matter, unlike Earth’s motion around the Sun. Weather lacks Bernoulli-shift regularity—the escaping information might be a butterfly’s wingbeat in the Amazon or a sneeze in Madagascar. Heisenberg in 1927 noted causality’s problem is less the law than the definition of “present.” Determinism once fed the certitudes of Newton’s, Laplace’s and Lagrange’s descendants; today knowledge has changed, recalling Eco’s Guillermo de Baskerville, for whom deciphering the world was detective work with only clues.
Chaotic systems demand new concepts. Describing an initial state by a phase-space region fails because Liouville’s theorem conserves volume, blocking any entropy function growing with time.
The Quantum Event and the Limits of Reversibility
The closing pages of the quantum-mechanics chapter press a hard claim: irreversible becoming marks every physical entity, inverting the classical denial of time’s arrow. If experience confirms prediction, no strictly reversible quantum system exists. The success of the traditional reversible description reflects not a fundamental truth but a peculiarity of the atom–electromagnetic-field interaction—the small value of the fine-structure constant. The atom is a stable structure only almost imperceptibly touched by irreversible becoming, which reflects our warm Universe, where stable structures can exist in isolation. In the hot early Universe of the first three hundred thousand years, and in high-energy collisions, physics should become a science of intrinsically irreversible processes.
Retrospectively, the notorious observation problem clarifies. Irreversibility, apparently introduced arbitrarily by the act of observation, is a necessary condition of knowledge. The unobservable character of the quantum world follows from the model role played by the notion of the integrable dynamical system. Classifying dynamical systems lets us identify that model’s limits and, following Karl Popper’s “metaphysical dream,” find a realism centered on the event rather than deterministic evolution—events that permit our experimental dialogue with the microscopic world. Bohr’s arguments on integrating the experimental device into the phenomenon mark the limits of the formalism based on integrable systems; their weakness was the absence of an intrinsic definition of what an experimental device is, leading to a regression that, as Von Neumann showed, ends in the observer’s consciousness. Now theory itself cuts this regression: the question is where the formalism ceases to be valid. The quantum event, characterized by a mean life rather than the “act of observation,” breaks the quantum superposition.
The authors stress openness and continuity, contingency without arbitrariness, planetary regularity against chaotic dice, and the warm Universe that permits atoms, molecules, and living beings. The universality of time’s arrow does not distance us from our world but reveals the coherence of what traditional physics opposed as “phenomenological” or “fundamental.” Time is the thread articulating descriptions of the Universe at every level; an “emergence” of irreversible time from an intemporal reality remains obscure.
The Birth of Time
Some twenty years before the book’s writing, Penzias and Wilson’s discovery of ~3°K photon radiation, in Wheeler’s words, confronted physics with its greatest crisis. This radiation, bathing the whole observable Universe, was recognized as the faint trace of the gigantic event constituting the Universe’s birth. The Big Bang hypothesis, until then mathematical speculation, became a physical problem. For the Greeks, “cosmos” implied that the totality of existence can only be thought as order, harmony, beauty—not “something” to deal with but a source of signification situating our existence. Christian thought tied the Universe inseparably to divine Creation. Giordano Bruno held the Universe thinkable only as the unthinkable, immobile, unalterable infinity, alien to the categories of finite creatures. At the end of the eighteenth century Kant made speculations about the Universe an illusion of reason: no concept of the Universe can exist since it cannot be an object of any experience. Hence some physicists’ rejection of a physics that makes the Universe a genuine object of science.
Many world views extrapolating from physical laws contravened the Kantian interdict: in the twentieth century a “Newtonian” Universe, eternal like matter and its laws, opposes a “thermodynamic” Universe, also eternal but evolving toward thermal death. Eternal return, born of this contrast, symbolizes both the scientific conflict between dynamics and thermodynamics and the ancient relation between the conception of the Universe and the meaning of our existence. For Abel Rey, eternal return derived from Poincaré’s recurrence theorem, which loses validity for unstable systems. For Nietzsche, it is less a consequence of science than an affirmation: the will to live each instant so as to bear its infinite repetition. In the last seventy years the alliance of theory and observation has profoundly transformed cosmological thought. Who would have imagined being led to situate the Universe’s thermal death not at the end of its history but at its origin? The idea that our Universe’s order is not a survivor of progressive degradation but an order produced by an original entropic explosion—whose gigantic cost the fossil radiation lets us evaluate—measures the ground covered.
Scenarios for the Universe’s birth constitute a laboratory of conceptual experimentation, scientifically legitimate because they create a new field testing the dialogue between theory and experience. That cosmology now forms part of physics does not mean it raises no conceptual problems. Start with the Big Bang itself: remounting the Universe’s evolution toward the past, we reach a singularity, a point without extension where all energy and matter are concentrated. The singularity has imposed itself thanks to the standard model’s success, yet neither this model nor physics generally lets us describe it.
The Birth of Time: Unification and the Primordial Universe
Einsteinian cosmology integrated space-time and matter but retained Newton’s definition of matter as masses and interactions, which is why the arrow of time went unnoticed there. That approximation is valid only once the genesis of the Universe is finished. The initial phase saw the coherent tissue of space-time torn, fusing geometry, expansion, and irreversibility. Unification has two aspects: the connection between geometry and dissipation, and the unification of interactions. Apart from the Big Bang singularity itself, the primordial Universe points to the quantum era, for which Cohen-Tannoudji and Spiro demanded a “third quantization” integrating particle physics with general relativity. This yields natural Planck units of length, time, and temperature—a Planck length, a Planck time of roughly 10⁻⁴⁴ seconds, a Planck temperature around 10³³ degrees. Until unified theories exist, models of the Universe’s first instants remain phenomenological.
The Planck mass suggests such monstrous particles are mini-black holes. Any massive body has a Schwarzschild radius; the Sun’s is one kilometer, Earth’s one centimeter. A particle of Planck mass would have a Schwarzschild radius equal to its Compton radius, defining the threshold at which elementary particles become mini-black holes. Hawking and Bekenstein showed black holes cannot definitively trap matter-energy: quantum uncertainty predicts a tunnel effect, making black holes thermodynamic objects with temperature and entropy; smaller mass means higher temperature, lower entropy, and shorter lifetime. A solar black hole would live 10⁶⁶ years; a Planck-mass one would vanish in Planck time. Primordial creatures would thus be essentially dissipative, with extremely high entropy. Only particles around fifty times the Planck mass—mini-black holes—could have torn the space-time tissue of the vacuum.
The scenario proposed by Brout, Englert, and Gunzig, continued by Gunzig and Nardone, starts from the Minkowski vacuum, defined quantum-mechanically: the quantum vacuum contains all possible particles in potency, with fluctuations not subject to energy conservation. If a virtual particle above a threshold appears, it triggers a strongly nonlinear cooperative mechanism—the “free lunch” theory—drawing positive energy from the negative energy of space-time’s geometry, curving space-time and materializing further particles. This resembles crystallization of a supercooled liquid via nucleation. The model shows exponential expansion at constant density, a de Sitterian inflationary phase, without a cosmological constant; production and inflation end with the disintegration of massive particles, during which nearly all the Universe’s entropy was produced. With their evaporation begins the adiabatic evolution of the standard model. There is no initial singularity, no Big Bang, but a matter-creating instability: any dissipative instability mechanism that creates matter suppresses the Big Bang—the Big Bang is structurally unstable.
Key Ideas
The principle of sufficient reason and dynamical reversibility. Leibniz’s principle, which guided the construction of dynamical languages from Galileo to Hamilton, equates cause and effect in a way that implies reversibility. A body descending an inclined plane loses height but gains velocity sufficient to reascend; Huyghens generalized this into the equality mgh = mv²/2. This principle describes more than determinism—it describes a symmetry that many causal processes, such as Fourier’s law of heat diffusion, contradict.
The H-theorem and molecular chaos. Boltzmann’s H function was constructed to decrease monotonically toward equilibrium, providing a microscopic model of irreversible evolution. But his derivation depended on the assumption of “molecular chaos”—that particle velocities are uncorrelated before collision. Loschmidt’s objection showed that velocity inversion maps every Boltzmannian initial state to an inverse-evolving one, and Boltzmann never satisfactorily justified the intrinsic difference between molecular chaos and correlated states.
Dissipative structures and self-organization. Far from equilibrium, irreversible processes become a source of coherence. Bénard vortices, chemical clocks, and autocatalytic reactions demonstrate that matter can spontaneously organize when driven sufficiently far from equilibrium. These “dissipative structures” are more costly in entropy than the unstable steady state they replace, and they exhibit a reciprocal causality in which the system’s activity gives meaning to external constraints like gravitation.
Chaotic attractors and the Lyapounov time. Chaotic systems have a temporal horizon set by the Lyapounov time, beyond which knowledge of the initial state loses pertinence. The baker’s transformation illustrates this: a reversible yet unstable evolution that fragments any region so nearby points diverge exponentially. Increasing precision to extend predictability becomes prohibitively costly—multiplying predictable time by ten requires increasing precision by a factor of e¹⁰.
The baker’s transformation and Bernouilli shifts. The baker’s transformation flattens a unit square into a rectangle and superposes the halves, with a dilatant horizontal coordinate and a contractant vertical one. Its dynamics correspond to Bernouilli shifts on binary decimals, where each decimal is independent and equiprobable like coin tosses. Periodic trajectories are the exception; ergodic trajectories cover phase space. This model shows how irreversibility emerges from a reversible law when finite precision is taken seriously.
Correlations and the arrow of time. Collisions create correlations between particles that escape control, forming a “sea” of incoherent correlations. Velocity inversion transforms post-collisional into pre-collisional correlations, but the arrow dominates: after more collisions, H recovers lower values. The correlation flow validates Boltzmann’s kinetic description and founds the probabilistic description rather than depending on it. Even in equilibrium, collisions create correlations that disappear almost immediately, so the past-future difference persists microscopically.
Hypnons. For denser systems, objects defined as independent in equilibrium required redefinition. “Hypnons” are entities that “sleep” in equilibrium while non-equilibrium awakens them and creates correlations between them. They were introduced via a non-local, time-symmetry-broken description derived from the baker model.
The quantum Zeno effect and Schrödinger’s cat. Misra and Sudarshan showed in 1977 that as the interval between repeated observations of an unstable system tends to zero, the quantum formalism concludes the atom cannot leave its excited level. This “Zeno paradox” completes what Schrödinger indicated with his cat: quantum mechanics cannot answer when an unstable particle has disintegrated, revealing the difficulties of the theory’s dualist nature.
The fine-structure constant and the warm Universe. The success of perturbative quantum mechanics stems from the weak electromagnetic interaction, measured by α = e²/hc = 1/137. If α were of order 1, perturbation theory would fail. The atom is a stable structure only almost imperceptibly marked by irreversible becoming, reflecting our “warm” Universe where stable structures can exist isolated. In the hot early Universe and in high-energy collisions, physics should become a science of intrinsically irreversible processes.
Planck units and mini-black holes. From c, G, and h one derives natural Planck units: a Planck length of roughly 10⁻³³ centimeters, a Planck time of roughly 10⁻⁴⁴ seconds, and a Planck temperature around 10³³ degrees. A particle of Planck mass (~10⁻⁵ grams) would have a Schwarzschild radius equal to its Compton radius, defining the threshold at which elementary particles become mini-black holes. Hawking and Bekenstein showed black holes are thermodynamic objects with temperature and entropy; a Planck-mass black hole would vanish in Planck time.
Themes
The book’s deepest theme is the cultural singularity of physics’ denial of time. How could a merely technical problem—the impossibility of defining a function that a dynamical system’s evolution makes grow with time—give rise to the almost inconceivable claim that the difference between past and future has no objective reality? No culture before the modern West had questioned such overwhelming evidence. The authors trace this to the metaphysical choice of determinism, embodied in references to a God who does not play dice, who knows simultaneously position and velocity, and to Laplace’s and Maxwell’s demons. Against this, they orient themselves toward Tagore’s conception of truth as a perpetual, open process of reconciliation between the universal human spirit and the spirit confined in each individual.
A second theme is the relationship between knowledge and the knower. Bohr’s meditation at Krónberg castle—that the same stones can speak of molecules, geological strata, fossils, or Hamlet’s doubts—expresses the inseparability of the problem of reality and human existence. The ideal that eliminates the describer and maintains reference to God has no alternative in a purely pragmatic conception. Our knowledge’s dependence on our relation to the world is not a limit but a source of new exigencies of coherence and pertinence.
A third theme is the inversion of the traditional hierarchy between “fundamental” and “phenomenological” descriptions. For most physicists after Boltzmann, the truth of physical time must be defined at the level of fundamental theories, from which the status of time in phenomenological descriptions derives. The authors’ path is inverse: they question the conception of physical time in fundamental theories starting from phenomenological evidence. This inversion raises the question of the authority historically conferred on these theories and leads to a new ideal of intelligibility.
A fourth theme is the constructive role of irreversibility. Far from being merely a source of disorder, irreversible processes are a source of coherence and organization. Snowflakes form regular spherical crystals near equilibrium but branched ones far from equilibrium; crystals are the memory of their formation path. Chemistry “creates its object,” as Berthelot said. The origin of life may lie at submarine hot springs where water at 275 atmospheres and 350°C meets cold ocean water—intense non-equilibrium. “In the beginning was activity”—life may be daughter of Earth before daughter of the Sun.
A fifth theme is the relationship between science and freedom. Leibniz showed why the ideal of divine knowledge could become inaccessible, illusory, and sterile: Buridan’s ass before two equally tempting meadows, Adam doubting whether to eat the forbidden apple—their choices cannot be predicted because prediction would require positively infinite knowledge. Any finite information defines a “diffuse” Adam compatible with infinite individual Adams capable of divergent destinies. Leibnizian freedom within a world governed by sufficient reason is not an illusion but a practical truth reflecting the distance only God can cross between being, which embraces the infinite, and our knowledge, finite by nature. The authors retrace this path regarding humbler physical objects, reaching the physical problem of irreversibility.
Why It Matters
The book matters because it challenges the image of physics as an almost imperialist enterprise imposing a single, timeless truth on all of reality. The authors hope readers will discover how foreign physical theories are to that image, revealing their character as bold, risky invention and creation of meaning. They explain how physics could deny time while phenomenological descriptions affirm it, and how it can now include time at the most fundamental level.
The stakes are not merely technical. If irreversibility is an irreducible physical concept, then the notion of “law of Nature” is transformed. The arrow of time’s universality does not distance us from our world but reveals the coherence of what traditional physics opposed as “phenomenological” or “fundamental.” Time becomes the thread articulating descriptions of the Universe at all levels. An “emergence” of irreversible time from an intemporal reality remains obscure, but the authors insist there is no turning back: the illegitimate ideal of intelligibility has been surpassed.
The book also matters for its method. It is not an introduction to contemporary physics—there is no general exposition of dynamics, thermodynamics, quantum mechanics, or relativity. Instead, it explores the confrontation between physics’ conceptual schemes and the problem of time, starting from Boltzmann’s work as the first act of this confrontation. The authors prefer to describe theoretical problems “live,” even before publication in specialized journals, making the book not a continuation of La nueva alianza but a more radical restatement of the problem of time.
Finally, the book matters because it takes seriously the possibility that the twentieth century is comparable only to the seventeenth. It began with relativity and quantum mechanics, yet these remain within the classical tradition regarding time. Unexpected experimental discoveries—elementary particles, non-equilibrium structures, the evolution of the Universe—all point toward overcoming the negation of irreversible time. A new coherence is perceived: at all levels of physics we find the time classical tradition denied. The authors’ wager is that this coherence will prove more fruitful than the old ideal of a timeless, deterministic world.
