THE DISORDERED MIND: LUDWIG BOLTZMANN
September 5, 1906
Duino, Italy
A fairly good-looking man in his early sixties, with an unkempt beard and piercing eyes glowering behind his thin-rimmed spectacles, stood at the edge of a fateful decision. As he sent his wife and daughter off for a swim during their holiday at the Bay of Duino, he silently prepared himself for what would be the last few hours of his life. With a mind as restless as the turbulent sea before him, he made his way toward a rope—his chosen instrument of departure. No farewell note, no final explanation. Just silence.
And with that, the chapter of one of history’s greatest scientists came to a close—a man whose mind was as chaotic as the very concept he helped define. Ironically, his most significant contribution to science was his work on entropy, the very measure of disorder itself.
Coincidence? Or a cruel twist of fate? We may never know...
BOLTZMANN: A PEEP INTO HIS LIFE
"The increase in disorder or entropy is what distinguishes the past from the future, giving a direction to time"
Stephen Hawking
Figure 1: Ludwig Boltzmann, the dis-ordered mind
As the son of a tax revenue officer, numbers may have been in his blood, but his restless mind was drawn to deeper questions. He grew up in the final decades of the Austro-Hungarian Empire, a time when the illusion of political stability was beginning to crack.
In 1869 Boltzmann was appointed to a chair of theoretical physics at Graz. He held this post for four years then, in 1873, he accepted the chair of mathematics at Vienna. He did not stay very long in any place and after three years he was back in Graz, this time in the chair of experimental physics.
Yet, while politics wavered, science surged forward. Boltzmann’s era was one of upheaval and discovery. Mathematician Georg Cantor challenged the very nature of infinity, while Charles Darwin’s Origin of Species shook the foundations of creationist thought. The world was expanding in ways few could have imagined.
His personality certainly had a major impact on the direction that his career took and personal relationships, where he was always very soft-hearted, played a big part. He suffered from an alternation of depressed moods with elevated, expansive or irritable moods. Indeed his physical appearance, being short and stout with curly hair, seemed to fit his personality. His fiancée called him her "sweet fat darling".Atoms, have you seen one yet?
Ernest Mach, intended for Boltzmann
During Boltzmann’s time, physics was undergoing a philosophical crisis, with different schools of thought competing to define the nature of reality. Two major opponents of Boltzmann’s atomism were Energism and Positivism—both rejecting the idea of invisible atoms and favoring alternative explanations of physical phenomena.
Figure 2: Boltzmann, Ostwald and Mach
Energism, led by Wilhelm Ostwald, argued that energy, not matter, was the fundamental entity of nature. According to this view, atoms were unnecessary constructs; instead, all physical phenomena could be explained purely in terms of energy transformations. Energists believed that energy alone was real, and matter was merely a convenient abstraction.
Ernest Mach and other positivists, which is basically a philosophical approach, dismissed theories that introduced "unobservable" entities—such as atoms—as metaphysical and unscientific. They argued that physics should only describe relationships between observable quantities, such as pressure, temperature, and volume, without assuming an underlying atomic structure.
SECOND LAW OF THERMODYNAMICS
Rudolf Clausius, one of the pioneers of thermodynamics, introduced the concept of entropy in 1865. He formulated the Second Law of Thermodynamics, stating that in any cyclic process, the total entropy of an isolated system can never decrease.
In the year 1865, Clausius summarized thermodynamics with:
1. The energy of the universe is constant. (First Law)
2. The entropy of the universe tends to a maximum. (Second Law)
This means that while energy is conserved, entropy always increases in a natural process.
Boltzmann asked a simple yet profound question—why does entropy never decrease? And the answer to this question changed the course of science and engineering forever!
-Failure of Newton's Laws:-
Isaac Newton's laws are time reversible meaning it doens't care if time moved forward or backward because the math always worked well either way.
For example: Imagine you are watching a video of two footballs colliding. If you play the video backward, it will still make sense—the footballs will just move in reverse.
Now imagine you are watching a video of a perfume bottle spraying, and the scent spreading.
If you play the video backward, it will look ridiculous- perfume molecules gathering back into the bottle.
In reality, we never see these things happen backward. But Newton’s laws don’t explain why!
Figure 4: The concept of disorderness couldn't be explained by Newton's Laws.
Boltzmann believed that if he could use Newton’s laws to track the motion of every single gas molecule, he could prove why gases naturally spread out over time.
Boltzmann thought that maybe Newton’s equations would show that molecules always tend to move from an ordered state (all in one corner) to a dis-ordered state (spread throughout the box).
Newton’s laws work both forward and backward in time. If you take a gas that has already spread out and reverse all the molecules' motions, they would reassemble into the corner again—which never happens in real life!
According to Newton’s equations, there was nothing stopping gas molecules from un-mixing themselves and gathering back together. But in reality, we never see this happen.
This meant that Newton’s laws alone couldn’t explain why gases always spread out and never the other way around.
-Maxwell steps in: the Maxwell-Boltzmann curve:-
So why do gases behave the way they do?
Boltzmann realized that the reason gases spread out isn’t a law of motion or a force—instead it was a matter of probability. There are trillions of ways for molecules to be randomly spread out, but only a few ways for them to all be neatly packed together. Since random motion is happening all the time, the system will almost always end up in the more probable (spread out) state.
Figure 5: James Clerk Maxwell
Earlier, the legendary scientist James Clerk Maxwell described a mathematical way to describe the distribution of speeds among gas molecules which could be explained with the help of the following example:
Imagine a room filled with gas molecules, each one too tiny to see, yet moving at incredible speeds, colliding, ricocheting, and constantly changing direction. Some molecules zipped through space like race cars, others lazily drifted, and many moved at an average speed-Maxwell immediately understood that maybe speeds of molecules weren’t uniform but followed a statistical distribution.
To explain his approach, Maxwell assumed the following conditions:
1. Molecules move randomly – There’s no preferred direction of motion.
2. Collisions are elastic – When molecules collide, they exchange energy, but no energy is lost.
3. The distribution of speeds should be the same in all directions – The gas doesn’t favor movement in any particular way.
Maxwell realized that since the gas molecules were moving in three dimensions (x, y, and z directions), their total speed v was related to their speeds in each direction as:
v=[v(x)]²+[v(y)]²+[v(z)]²
From this expression, he obtained the following equation shows how the speeds of particles in a gas are distributed at a given temperature, with the x-axis representing speed and the y-axis representing the number of particles:
Interpretation:
4. Effect of Temperature on the Curve:
‣Higher Temperature: The curve spreads out, shifting to the right, meaning more molecules move faster.
‣Lower Temperature: The curve narrows and shifts to the left, meaning most molecules move more slowly.
BOLTZMANN'S ENTROPY FORMULA
"Bring forward what is true, write it so that it is clear, defend it to your last breath!"
Ludwig Boltzmann
Maxwell’s idea suggested that instead of treating each individual molecule with exact equations (which would be impossible for billions of particles), one could describe the system as a whole using statistics.
Boltzmann took Maxwell’s idea and pushed it further, according to him:
1. A gas is made of a huge number of molecules, each moving in random directions and colliding.
2. Instead of tracking each molecule, we could count the number of ways these molecules could arrange themselves.
3. The more possible arrangements (microstates) that correspond to a given overall state (macrostate), the more likely it is to occur.
He formalised this insight with his famous entropy equation:
S=k×ln(W)
where,
S→ entropy
k→Boltzmann constant
W→ no of possible microstates corresponding to a given microscopic state
This equation means that entropy increases not because of some fundamental force, but simply because disordered states are far more probable than ordered ones.
To grasp this intuitively, imagine rolling two dice. The probability of getting double sixes (6,6) is just 1 in 36. If you shake the dice repeatedly, you're far more likely to get random numbers than another perfect pair of sixes.
Now, imagine rolling millions of dice—each representing an individual molecule. The chance that they all land on six is so incredibly small that it practically never happens. Similarly, in a gas or any physical system, the most ordered states are the least probable, and as a result, a system will always move toward a more probable, disordered state.
CONCLUSION:
Despite his struggles, history would prove him right. His famous equation, which elegantly connects entropy to microscopic probabilities, became one of the cornerstones of statistical mechanics.
🤍
ReplyDeleteThank you Vishal! 😊
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