Entropy Across Worlds – One Measure, Many Stories

How a Word Escaped the Lab

There is a peculiar habit in science of taking a word defined with rigid mathematical precision in one room and letting it drift into the hallway, where it is picked up by neighbors who use it to describe something entirely different. “Energy” suffered this fate long ago. “Quantum” is currently undergoing it. But perhaps no term has traveled further from its origin while retaining such an aura of fundamental truth as entropy.

It began as a measure of waste in steam engines—a thermodynamic accounting of heat that could no longer do work. Then, through the statistical mechanics of Boltzmann, it became a matter of counting: how many microscopic arrangements of atoms could pass for the same macroscopic object? In my own work at Bell Labs, we found that this same logarithmic measure, $H = -\sum p_i \log p_i$, governed not just heat, but information—the resolution of uncertainty in a message.

Today, the signal has propagated even further. We see “entropy” invoked to explain the “dark matter” inside artificial intelligence models, where features exist in superposition, hidden from our view. We hear it in the study of consciousness, where the “noise” of mental chatter is contrasted with the low-entropy signal of pure awareness.

The question before us is one of signal integrity. When we trace this concept across these three distinct worlds—the physical arrow of time, the hidden geometry of neural networks, and the phenomenology of the mind—are we measuring the same underlying quantity? Or have we merely found a metaphor that is convenient, evocative, and mathematically loose? To answer this, we must look at the channel capacity of the idea itself.

Microstates and the Arrow

In the physical world, entropy is not merely a description of disorder; it is a counting exercise. As the work of ScienceClic elucidates, entropy is fundamentally about configuration counting. Imagine a gas in a box. There are vastly more ways for the molecules to be scattered randomly throughout the container than there are for them to be huddled in one corner. If you are blind to the precise position of every atom—if you are observing the system from a “coarse-grained” macroscopic perspective—the scattered state is simply overwhelmingly more probable.

This probability gradient creates what we perceive as the arrow of time. We do not see shattered teacups reassemble not because the laws of physics forbid it, but because the number of microstates corresponding to “shattered” dwarfs the number corresponding to “whole.” The system wanders into the larger phase space simply because there is more of it.

This leads to the ultimate extrapolation: the heat death of the universe. If the cosmos is an isolated system, it must drift toward the configuration with the highest possible entropy—a state of perfect equilibrium where energy is uniformly distributed, gradients vanish, and no work can be done. It is the ultimate noise. A message that has become all static.

However, this thermodynamic view relies heavily on the observer’s inability to distinguish microstates. It is a measure of our ignorance as much as the system’s state. If we could track every particle, the “entropy” would be constant. It is only because we compress the data—grouping trillions of atomic positions into a single variable called “temperature” or “pressure”—that entropy emerges as a meaningful quantity. It is the cost of compression.

The Dark Matter of Intelligence

When we turn to the artificial minds we are currently building, we find a striking parallel. In the realm of mechanistic interpretability, as explored by Welch Labs, researchers are grappling with a form of information entropy that defines the limits of what we can understand about our own creations.

Modern language models are vast, high-dimensional spaces. We might naively assume that one neuron equals one concept—a “grandmother neuron” for the idea of a grandmother. But the reality is far more efficient, and far more chaotic. Through a phenomenon known as the superposition hypothesis, models appear to pack more concepts into their layers than they have neurons to represent them. They do this by encoding features as linear combinations of neurons, utilizing the high-dimensional geometry to store information in a compressed, “polysemantic” state.

This is an entropy problem. The model has maximized its channel capacity, transmitting more “meaning” than the physical channel (the number of neurons) would seemingly allow. For the interpretability researcher, this creates a “dark matter” problem. We can probe the model, but the features we extract—using tools like sparse autoencoders—account for less than 1% of the model’s total knowledge. The rest is hidden in the noise, in the interference patterns between neurons.

Here, entropy is not about heat death, but about encryption. The model has learned an encoding scheme that is optimal for its objective function but opaque to us. We are like the thermodynamic observer who sees only temperature; we see the model’s output, but the rich microstates of its internal reasoning are lost to our coarse-grained analysis. The “dark matter” is the information we cannot yet decompress.

When Minds Feel Entropy

The signal becomes noisier still when we enter the domain of subjective experience. In the explorations of Mountain Consciousness, we encounter descriptions of the mind that sound suspiciously thermodynamic. We hear of the “thoughtless thinker,” a state where the ceaseless “chatter” of the mind—the high-entropy noise of anxiety, planning, and memory—subsides into a state of “liquid logic” or pure awareness.

Is this entropy? Or is it just a mood?

There is a structural argument to be made. The ordinary waking state is characterized by high informational entropy: the attention flits rapidly between objects, thoughts, and sensations. The “configuration space” of the mind is vast and chaotic. In contrast, the state of “awareness independent of thought flow”—the witness consciousness—is described as a “field phenomenon” rather than a localized spark. It is state-invariant. In information theory terms, it is a signal with zero bandwidth, or perhaps infinite redundancy. It carries no specific message, and therefore has low entropy.

The “liquid logic” described—a mode of thought that flows, loops, and adapts without crystallizing into rigid binaries—resembles a system operating at the “edge of chaos,” a phase transition where information processing is maximized. It is neither the frozen order of rigid dogma (low entropy, low information) nor the random noise of psychosis (high entropy, no meaning). It is the complex middle ground.

However, we must be careful. The “entropy” of a meditator’s brain waves (measurable via EEG) is a physical quantity. The “entropy” of their subjective experience is a metaphor. When a yogi speaks of “surrendering the need for truth,” they are describing a psychological release, not a thermodynamic one. Yet, the parallel holds a certain resonance: both the steam engine and the anxious mind suffer from an excess of useless motion.

One Measure or Many?

So, do we have one measure, or three?

Mathematically, the link between Boltzmann’s entropy and my own definition of information entropy is solid. They are the same formula. A system with high thermodynamic entropy requires more bits to describe its microstate. The “dark matter” of AI is a direct application of this: the model’s internal state has high information content that we are struggling to decode.

The bridge to consciousness is more precarious. It relies on a structural analogy—noise vs. signal, chaos vs. order—rather than a countable quantity. We cannot calculate the bits required to encode a “thoughtless” state of mind because we do not yet know the code of the mind itself.

And yet, the intuition persists for a reason. In all three worlds, entropy is the shadow of limitation.

• In physics, it is the limit of work we can extract from heat.
• In AI, it is the limit of meaning we can extract from a compressed representation.
• In the mind, it is the limit of clarity we can achieve amidst the noise of cognition.

We are finite observers in an infinite universe. Entropy is simply the name we give to the parts of reality that exceed our bandwidth. Whether it is the random motion of a gas, the superposed features of a neural network, or the turbulent stream of consciousness, entropy is the measure of what we cannot control, what we cannot predict, and what we must ultimately—whether through mathematics or meditation—accept.

The signal is clear: we do not eliminate the noise. We only learn to navigate it.