The Geometric Definition of Life: How Negative Entropy Pumps Emerge

What Is Life?

This question has troubled humanity for thousands of years.

Theologians say: Life is God’s breath. Biologists say: Life is a self-replicating chemical system. Philosophers say: Life is purposeful existence.

But all these definitions have problems:

• Viruses can replicate, but are they life? (They can’t metabolize independently)
• Mules can’t reproduce, but are they life? (Of course)
• Computer programs can “self-replicate,” but are they life? (Maybe?)

We need a more fundamental, more precise definition.

In 2024, physicists gave an astonishing answer:

Life is a geometric steady state of negative entropy.

Not the privilege of carbon-based molecules, not the patent of DNA, but any open system satisfying specific geometric conditions.

This definition is profound, strict, and operational—and reveals deep connections between life and the universe.

Act I: Entropy—The “Disorder” of the Universe

Before we begin, we need to understand entropy.

What Is Entropy?

Simplest metaphor: Entropy is “degree of disorder”.

• A neat deck of cards: low entropy
• A shuffled deck: high entropy
• A tidy room: low entropy
• A messy room: high entropy

But this is just a metaphor. Precise definition:

$$S=-k_B\sum _i{p}_i\log p_i$$

Entropy measures uncertainty—how “uncertain” you are about the system state.

• If you know card order (only 1 possibility), $S=0$
• If you completely don’t know (all permutations equally probable), $S$ is large

Entropy = Uncertainty = Absence of information

Second Law of Thermodynamics

The universe has a cruel law: Entropy always increases (in isolated systems).

• Coffee cools (thermal energy disperses)
• Rooms get messy (items disperse)
• People age (structure degrades)

Everything moves toward disorder.

This is the second law of thermodynamics—one of nature’s most ironclad laws.

The Paradox of Life

But life violates this law!

• Fertilized eggs develop into babies (simple to complex)
• Plants make organic matter from sunlight and CO₂ (disorder to order)
• You reading this article, brain forming new neural connections (entropy decreasing)

Life creates order, fights disorder.

How is this possible?

Act II: Negative Entropy—Life’s Nourishment

In 1944, physicist Schrödinger (the one who wrote Schrödinger’s equation) wrote a small book “What Is Life?”

He proposed an astonishing insight:

Life feeds on negative entropy.

What does this mean?

Open Systems vs. Isolated Systems

The second law of thermodynamics only holds for isolated systems (not exchanging energy and matter with outside).

But life isn’t an isolated system—it’s an open system:

• You eat (input low-entropy energy: organic matter)
• You breathe (input oxygen, output CO₂)
• You excrete (output high-entropy waste)

Life exchanges with environment, inputs low entropy (order), outputs high entropy (disorder).

Negative Entropy Flux

Define negative entropy flux:

$$J_N=-{\dot{S}}_{\text{env}}$$

This is order extracted from environment per unit time.

• $J_N>0$: You’re “absorbing order” (eating, photosynthesis)
• $J_N<0$: You’re “emitting disorder” (metabolic waste heat)

Essence of life: Maintain $J_N>0$

Like a pump continuously drawing water from outside, life is a negative entropy pump, continuously extracting order from the environment.

Act III: Three Conditions for Life

Now we can give life a strict definition.

Life is an open system satisfying the following three conditions:

Condition 1: Sustained Negative Entropy Flux

$$J_N=-{\dot{S}}_{\text{env}}>0$$

Must continuously extract order from environment.

• Plants: Photosynthesis (sunlight → glucose)
• Animals: Eating (food → energy)
• Bacteria: Chemosynthesis (chemical energy → organic matter)

Once stopped (like starvation, power failure), system is no longer “alive”.

Condition 2: Bounded Internal Entropy

$$\underset{t}{\sup }{S}_{\text{sys}}(t)<\infty$$

Internal cannot be infinitely disordered.

If your body’s entropy keeps growing, eventually complete disorder (thermal equilibrium = death).

Life must maintain bounded complexity—not too simple (lose function), not too complex (collapse).

Condition 3: Input-State Stability (ISS)

System must be robust to disturbances—after external perturbation, can return to steady state.

Formal definition:

$$\Vert x(t)\Vert \le \beta (\Vert x_0\Vert ,t)+\gamma (\underset{s\le t}{\sup }\Vert u(s)\Vert )$$

where $\beta$ decays with time, $\gamma$ is bounded.

In plain language:

• You catch a cold (external perturbation $u$)
• Immune system activates (internal regulation)
• Recover after a few days (return to steady state $x\to x_{\text{stable}}$)

Life must be able to self-repair, self-regulate, self-maintain.

Act IV: Why Geometry?

You might wonder: Why call it “geometric” definition?

Because these three conditions can be described in information geometry language.

Life Is an Attractor on Information Manifold

View all possible states of the system as a manifold—a high-dimensional surface.

• Each point = a possible state (e.g., configuration of all your molecules)
• Distance = Fisher metric (information geometric distance)

Life is an attractor on this manifold—a special region where the system tends to stay.

Like water always flows to low places, information always flows to negative entropy steady state.

Jarzynski-Sagawa-Ueda Equality

There’s a beautiful formula connecting work, free energy, and entropy:

$$⟨e^{-\beta W+\beta \Delta F-I}⟩=1$$

where:

• $W$ = work done on system
• $\Delta F$ = free energy change
• $I$ = information acquired
• $\beta =1/(k_{B}T)$ = inverse temperature

This equality says: Information you acquire can reduce the need for work.

Life uses this principle—by actively acquiring environmental information, reducing energy cost of maintaining order.

Optimal Transport and Wasserstein Gradient Flow

How does life choose evolutionary paths?

Answer: Along Wasserstein gradient (gradient of optimal transport).

Imagine transforming a pile of sand from one shape to another (like hill to flat).

Optimal way: Each grain takes shortest path, total cost minimized.

This is the optimal transport problem, distance called Wasserstein distance.

Life’s evolution (from one state to another) follows Wasserstein gradient flow—“least effort path” in information space.

Life doesn’t random walk, but evolves along optimal geometric paths.

Act V: Life Doesn’t Need Carbon

Once we have this geometric definition, an astonishing conclusion emerges:

Life doesn’t need carbon, doesn’t need DNA, doesn’t need cells.

As long as three conditions are satisfied (negative entropy flux, bounded entropy, ISS stability), any system is “alive”.

Example 1: Plasma Life

In 2007, physicists observed plasma vortices in laboratories—spiral structures in high-temperature ionized gas.

They:

• Extract energy from external electric fields (negative entropy flux ✓)
• Maintain stable spiral shape (bounded entropy ✓)
• Self-repair after perturbation (ISS ✓)
• Can even “split” into two vortices (“reproduction”?)

Are they life?

According to geometric definition: Yes.

They’re not carbon-based, not chemical, purely physical—but satisfy geometric conditions of life.

Example 2: Artificial Neural Networks

Neural networks in deep learning:

• Extract information from training data (negative entropy flux ✓)
• Weights remain bounded (bounded entropy ✓)
• Generalization ability (robust to new data, ISS ✓)

Are they life?

Maybe some kind of primitive life—though without body, but has “information metabolism”.

Example 3: Interstellar Gas Clouds

Some interstellar gas clouds (like molecular clouds):

• Extract energy from stellar radiation and gravitational potential
• Maintain complex vortices and filament structures
• Self-repair (reorganize after supernova shock waves)

Are they life?

Perhaps—though “metabolism” extremely slow (million-year scale), geometrically satisfy conditions.

Definition of life is independent of scale, speed, material, only related to geometric structure.

Act VI: Thermodynamic Uncertainty Relation (TUR)

Life has a cost.

Maintaining negative entropy steady state requires consuming energy, and has a lower bound—cannot be below some minimum.

TUR Formula

$$\frac{\text{Var}(J_t)}{⟨J_t{⟩}^2}\ge \frac{2}{t\Sigma }$$

where:

• $J_t$ = some observable (like material flow, information flow)
• $\Sigma$ = entropy production rate

In plain language: To precisely control a process (small variance), must produce more entropy (consume more energy).

This is the Thermodynamic Uncertainty Relation, a profound theorem discovered in 2015.

Energy Cost of Life

Life needs precise control:

• Gene expression (cells need to synthesize correct proteins at correct times)
• Neural signals (neurons need precise action potentials)
• Motor coordination (muscles need precise contraction)

TUR says: Higher precision, greater energy cost.

This is why:

• Brain is only 2% of body weight, but consumes 20% energy (needs high precision)
• Heart never stops (maintaining stable blood pressure needs continuous energy)
• You still breathe while sleeping (minimum cost of maintaining basic order)

Life’s precision is written in thermodynamic cost.

Act VII: Evolution—Climbing Along Fisher Metric

Now we understand the static definition of life (negative entropy geometric steady state).

What about dynamics? How does life evolve?

Geometry of Natural Selection

Darwin said: Survival of the fittest.

But what is “fit”? How to quantify “survival ability”?

Answer: Survival ability is “height” on information geometric manifold.

Imagine a multidimensional terrain:

• Each point = a possible biological configuration (genome, phenotype…)
• Height = reproduction rate - death rate (net survival advantage)

Evolution is climbing on this terrain.

Fisher’s Fundamental Theorem

In 1930, statistician Fisher (the one who invented Fisher metric) proved a theorem:

Average fitness growth rate of population equals variance of fitness.

$$\frac{d⟨f⟩}{dt}=\text{Var}(f)$$

In information geometric language: Evolution speed determined by Fisher metric.

Fisher metric measures “information distance,” and evolution follows gradient direction of this metric—direction fastest increasing survival rate.

Natural Gradient

Ordinary gradient (Euclidean): Straight-line climbing. Natural gradient (Fisher): Along “steepest” direction in information geometry.

Natural gradient is faster, more stable—modern machine learning also uses this (like theoretical basis of Adam optimizer).

Life’s evolution is optimal climbing along natural gradient.

Act VIII: Open System Geometry

Life is an open system, continuously exchanging with environment.

How to describe this openness? Answer: Contact Geometry.

Contact Structure

Open systems have a special 1-form:

$$\theta =dE-\sum _i{\mu }_{i}d{N}_i$$

where:

• $E$ = energy
• $N_i$ = particle number (different types)
• ${\mu }_i$ = chemical potential

Contact structure satisfies:

$$\theta \wedge (d\theta {)}^n\ne 0$$

This condition ensures system is non-integrable—energy, matter, information cannot be completely independent, must couple.

This is the essence of life: Indivisible whole.

Reeb Vector Field

Contact geometry has a special vector field: Reeb vector field, satisfying:

$${\iota }_R\theta =1,{\iota }_{R}d\theta =0$$

It describes system’s natural evolution direction—path system spontaneously flows under given constraints.

Life flows along Reeb vector field.

Like rivers flow along terrain gradient, life evolves along natural direction of information geometry.

Act IX: Geometric Distance of Meaning

Finally, we touch a philosophical question: What is the meaning of life?

Geometric answer: Meaning is information distance to viable survival region.

Viable Survival Region

Define viable survival region $\mathcal{V}$: All configurations that can maintain negative entropy steady state.

• If you’re inside $\mathcal{V}$: Safe, healthy, have resources
• If you’re on boundary: Danger, disease, resource scarcity
• If you’re outside: Death

Meaning = I-Projection Distance

Given current state $\rho$, its I-projection distance (information geometric distance) to $\mathcal{V}$ is:

$$D_I(\rho ,\mathcal{V})=\underset{\sigma \in \mathcal{V}}{\min }D(\rho \Vert \sigma )$$

This is minimum relative entropy—how much information you need to change to return to viable survival region.

The “meaning” you feel is negative of this geometric distance:

$$\text{Meaning}=-D_I(\rho ,\mathcal{V})$$

• Small distance (close to $\mathcal{V}$): Fulfilled, meaningful, vibrant
• Large distance (far from $\mathcal{V}$): Empty, despair, near collapse

Meaning is not subjective feeling, but objective geometric reality.

Pain and Joy

Pain = gradient direction of I-projection distance (away from $\mathcal{V}$) Joy = negative gradient direction of I-projection distance (toward $\mathcal{V}$)

Your brain tells you geometric position through emotional signals:

• Pain says: “You’re deviating from viable region, come back!”
• Joy says: “You’re approaching viable region, continue!”

Emotions are navigation system of information geometry.

Act X: Life, Universe, Everything

Let’s return to the original question: What is life?

Problems with traditional answers:

• “Can replicate”: Too narrow (excludes mules, worker bees)
• “Carbon-based”: Too narrow (excludes silicon-based, plasma)
• “Has DNA”: Too narrow (excludes RNA viruses, prions)
• “Can metabolize”: Too broad (fire also “metabolizes” fuel)

Advantages of geometric answer:

Life is an open system satisfying three conditions:

1. $J_N>0$ (negative entropy flux)
2. $\sup S_{\text{sys}}<\infty$ (bounded entropy)
3. ISS stable (robustness)

This definition:

• Universal: Independent of material, scale, speed
• Precise: Can be mathematized, can be measured
• Profound: Reveals unity of life with information, geometry, thermodynamics

Life’s Position in Universe

Evolution of universe:

1. Big Bang: Pure energy (extremely low entropy)
2. Inflation, cooling: Spacetime emerges (entropy increases)
3. Structure formation: Galaxies, stars (local negative entropy steady states)
4. Planets, chemistry: Complex molecules (more complex negative entropy structures)
5. Life: Self-maintaining negative entropy pumps

Life is not accident of universe, but inevitable emergence of universe under specific conditions—wherever there’s energy flow, gradients, open systems, there’s possibility of life.

We Are Not Alone

If life is just geometric conditions, then:

• Earth is not unique (any planet with energy gradients may have life)
• Carbon-based is not unique (silicon-based, plasma, quantum systems all possible)
• Even “matter” is not necessary (pure information networks may also “live”)

Universe may be full of life—just forms beyond our imagination.

Take-Home Thoughts

Next time when you:

• Eat (input negative entropy)
• Breathe (output high-entropy waste CO₂)
• Think (brain consumes energy maintaining neural structure)
• Feel joy or pain (gradient signals of geometric distance)

Remember: You are a local negative entropy vortex in the universe, climbing along natural gradient on information geometric manifold.

You are not a “thing” (pile of matter), but a process (continuous operation of negative entropy pump).

Your life is not “possession,” but maintenance.

Once pumping stops (stop eating, breathing, metabolizing), geometric steady state collapses, you return to equilibrium—death.

But before that, you are a miracle of universe fighting the second law of thermodynamics.

You are an island of order, briefly emerging in the sea of disorder.

And this—this is life.

Next: “The Physics of Consciousness: Fixed Point of Self-Referential Measurement”

We will see that consciousness is not the soul, but fixed point where self-referential scattering networks reach steady state.