Building a Raspberry Pi Compute Cluster in the era of AI
Published:
This is not another tutorial on how to build a Raspberry Pi SLURM Cluster
Originally published on Medium.
Posts by Tags
AI
Collision Integrators
Fast spectral Boltzmann solvers: where they shine, where they stop, and what industry actually needs
Published:
The Boltzmann collision operator is one of the most expensive pieces of kinetic theory. In three velocity dimensions, a naive quadrature costs on the order of (N^6) operations per evaluation, where (N) is the resolution in each direction. For decades, that cost made full deterministic kinetic simulations impractical outside academic benchmarks. Fast spectral methods changed the picture—not by simplifying the physics, but by exploiting the convolution structure hidden inside the collision integral.
Compute Cluster
cluster-smi: CLI tool for monitoring Raspberry Pi clusters
Published:
When you run a rack full of Raspberry Pis, the question that comes up constantly is not “which process is on which CPU,” but something simpler:
How is my cluster’s hardware doing right now?
Building a Raspberry Pi Compute Cluster in the era of AI
Published:
This is not another tutorial on how to build a Raspberry Pi SLURM Cluster
Originally published on Medium.
Fast Fourier Transforms
Fast spectral Boltzmann solvers: where they shine, where they stop, and what industry actually needs
Published:
The Boltzmann collision operator is one of the most expensive pieces of kinetic theory. In three velocity dimensions, a naive quadrature costs on the order of (N^6) operations per evaluation, where (N) is the resolution in each direction. For decades, that cost made full deterministic kinetic simulations impractical outside academic benchmarks. Fast spectral methods changed the picture—not by simplifying the physics, but by exploiting the convolution structure hidden inside the collision integral.
Hydrodynamics
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 15
Published:
This is the right place to bring the journey to a deeper level.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 14
Published:
We have reached the point where the formal machinery meets real quantum matter. Up to now we have built a hierarchy: \(\text{Boltzmann} \rightarrow \text{BGK} \rightarrow \text{Chapman–Enskog} \rightarrow \text{quantum transport} \rightarrow \text{quantum hydrodynamics}.\)
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 13
Published:
This is the point where the two threads we have developed—quantum kinetic theory and wave-mechanical quantum hydrodynamics—are brought together.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 12
Published:
This chapter takes us into a subtle but very important distinction in quantum fluid theory.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 11
Published:
This is where the kinetic derivation connects with modern quantum fluids. Up to now we have treated quantum statistics as a modification of the equilibrium distribution. In this chapter we will explore what happens in the strongly quantum regimes, where the gas is no longer a small correction to the classical picture.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 10
Published:
This is a very good point in the derivation to stop and compare the three theories side by side, because the underlying architecture is now visible.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 9
Published:
We now arrive at the final closure step: extracting the quantum viscous stress tensor and quantum heat flux from \(f_q^{(1)}\).
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 8
Published:
We now proceed to repeat the Chapter 5 calculation for \(f_q^{eq}\) and derive the explicit quantum \(f_q^{(1)}\).
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 7
Published:
We have now completed the classical derivation up to the Navier–Stokes closure. This gives us a very useful reference point, because the quantum derivation will not be a completely new calculation—it will be a deformation of the classical calculation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 6
Published:
Most books state identities like \(\int c_i c_j c_k c_l e^{-\beta c^2}\,d^3c = A(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\) without proving them. That is unfortunate, because these tensor identities are the entire mathematical engine behind the Navier–Stokes constitutive laws.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 5
Published:
We have now arrived at what I consider the centerpiece of the entire Chapman–Enskog derivation. From this point onward, every line of algebra has a direct physical interpretation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 4
Published:
To derive the Navier–Stokes equations from the Boltzmann equation, we need to introduce the Chapman–Enskog expansion. Let’s begin.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 3
Published:
Having derived the classical Maxwell–Boltzmann equilibrium distribution and its moments, let us now derive the macroscopic conservation laws from the Boltzmann equation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 2
Published:
Let us now derive the classical Maxwell–Boltzmann equilibrium distribution from first principles (rather than simply stating it), prove why it is isotropic in the peculiar velocity, evaluate all of its velocity moments explicitly using Gaussian integrals, and show how the ideal-gas equation of state emerges directly from kinetic theory.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 1
Published:
Since I was a PhD student in kinetic theory, one question that got stuck in my mind was:
How does a microscopic description forget information and become a macroscopic fluid?
Kinetic Theory
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 15
Published:
This is the right place to bring the journey to a deeper level.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 14
Published:
We have reached the point where the formal machinery meets real quantum matter. Up to now we have built a hierarchy: \(\text{Boltzmann} \rightarrow \text{BGK} \rightarrow \text{Chapman–Enskog} \rightarrow \text{quantum transport} \rightarrow \text{quantum hydrodynamics}.\)
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 13
Published:
This is the point where the two threads we have developed—quantum kinetic theory and wave-mechanical quantum hydrodynamics—are brought together.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 12
Published:
This chapter takes us into a subtle but very important distinction in quantum fluid theory.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 11
Published:
This is where the kinetic derivation connects with modern quantum fluids. Up to now we have treated quantum statistics as a modification of the equilibrium distribution. In this chapter we will explore what happens in the strongly quantum regimes, where the gas is no longer a small correction to the classical picture.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 10
Published:
This is a very good point in the derivation to stop and compare the three theories side by side, because the underlying architecture is now visible.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 9
Published:
We now arrive at the final closure step: extracting the quantum viscous stress tensor and quantum heat flux from \(f_q^{(1)}\).
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 8
Published:
We now proceed to repeat the Chapter 5 calculation for \(f_q^{eq}\) and derive the explicit quantum \(f_q^{(1)}\).
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 7
Published:
We have now completed the classical derivation up to the Navier–Stokes closure. This gives us a very useful reference point, because the quantum derivation will not be a completely new calculation—it will be a deformation of the classical calculation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 6
Published:
Most books state identities like \(\int c_i c_j c_k c_l e^{-\beta c^2}\,d^3c = A(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\) without proving them. That is unfortunate, because these tensor identities are the entire mathematical engine behind the Navier–Stokes constitutive laws.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 5
Published:
We have now arrived at what I consider the centerpiece of the entire Chapman–Enskog derivation. From this point onward, every line of algebra has a direct physical interpretation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 4
Published:
To derive the Navier–Stokes equations from the Boltzmann equation, we need to introduce the Chapman–Enskog expansion. Let’s begin.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 3
Published:
Having derived the classical Maxwell–Boltzmann equilibrium distribution and its moments, let us now derive the macroscopic conservation laws from the Boltzmann equation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 2
Published:
Let us now derive the classical Maxwell–Boltzmann equilibrium distribution from first principles (rather than simply stating it), prove why it is isotropic in the peculiar velocity, evaluate all of its velocity moments explicitly using Gaussian integrals, and show how the ideal-gas equation of state emerges directly from kinetic theory.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 1
Published:
Since I was a PhD student in kinetic theory, one question that got stuck in my mind was:
How does a microscopic description forget information and become a macroscopic fluid?
Monitoring
cluster-smi: CLI tool for monitoring Raspberry Pi clusters
Published:
When you run a rack full of Raspberry Pis, the question that comes up constantly is not “which process is on which CPU,” but something simpler:
How is my cluster’s hardware doing right now?
Quantum Fluids
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 15
Published:
This is the right place to bring the journey to a deeper level.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 14
Published:
We have reached the point where the formal machinery meets real quantum matter. Up to now we have built a hierarchy: \(\text{Boltzmann} \rightarrow \text{BGK} \rightarrow \text{Chapman–Enskog} \rightarrow \text{quantum transport} \rightarrow \text{quantum hydrodynamics}.\)
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 13
Published:
This is the point where the two threads we have developed—quantum kinetic theory and wave-mechanical quantum hydrodynamics—are brought together.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 12
Published:
This chapter takes us into a subtle but very important distinction in quantum fluid theory.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 11
Published:
This is where the kinetic derivation connects with modern quantum fluids. Up to now we have treated quantum statistics as a modification of the equilibrium distribution. In this chapter we will explore what happens in the strongly quantum regimes, where the gas is no longer a small correction to the classical picture.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 10
Published:
This is a very good point in the derivation to stop and compare the three theories side by side, because the underlying architecture is now visible.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 9
Published:
We now arrive at the final closure step: extracting the quantum viscous stress tensor and quantum heat flux from \(f_q^{(1)}\).
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 8
Published:
We now proceed to repeat the Chapter 5 calculation for \(f_q^{eq}\) and derive the explicit quantum \(f_q^{(1)}\).
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 7
Published:
We have now completed the classical derivation up to the Navier–Stokes closure. This gives us a very useful reference point, because the quantum derivation will not be a completely new calculation—it will be a deformation of the classical calculation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 6
Published:
Most books state identities like \(\int c_i c_j c_k c_l e^{-\beta c^2}\,d^3c = A(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\) without proving them. That is unfortunate, because these tensor identities are the entire mathematical engine behind the Navier–Stokes constitutive laws.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 5
Published:
We have now arrived at what I consider the centerpiece of the entire Chapman–Enskog derivation. From this point onward, every line of algebra has a direct physical interpretation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 4
Published:
To derive the Navier–Stokes equations from the Boltzmann equation, we need to introduce the Chapman–Enskog expansion. Let’s begin.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 3
Published:
Having derived the classical Maxwell–Boltzmann equilibrium distribution and its moments, let us now derive the macroscopic conservation laws from the Boltzmann equation.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 2
Published:
Let us now derive the classical Maxwell–Boltzmann equilibrium distribution from first principles (rather than simply stating it), prove why it is isotropic in the peculiar velocity, evaluate all of its velocity moments explicitly using Gaussian integrals, and show how the ideal-gas equation of state emerges directly from kinetic theory.
From Kinetic Theory to Quantum Fluids: A Guided Journey Through the Emergence of Hydrodynamics – Chapter 1
Published:
Since I was a PhD student in kinetic theory, one question that got stuck in my mind was:
How does a microscopic description forget information and become a macroscopic fluid?
Raspberry Pi
cluster-smi: CLI tool for monitoring Raspberry Pi clusters
Published:
When you run a rack full of Raspberry Pis, the question that comes up constantly is not “which process is on which CPU,” but something simpler:
How is my cluster’s hardware doing right now?
Building a Raspberry Pi Compute Cluster in the era of AI
Published:
This is not another tutorial on how to build a Raspberry Pi SLURM Cluster
Originally published on Medium.
Spectral Boltzmann Solvers
Fast spectral Boltzmann solvers: where they shine, where they stop, and what industry actually needs
Published:
The Boltzmann collision operator is one of the most expensive pieces of kinetic theory. In three velocity dimensions, a naive quadrature costs on the order of (N^6) operations per evaluation, where (N) is the resolution in each direction. For decades, that cost made full deterministic kinetic simulations impractical outside academic benchmarks. Fast spectral methods changed the picture—not by simplifying the physics, but by exploiting the convolution structure hidden inside the collision integral.