114 Scientific Diagrams

2d Convolution

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A two-dimensional convolution operator slides the kernel matrix across the target image and records elementwise products. Makes heavy use of the matrix environment in TikZ.

Autoencoder

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Basis + Lattice

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Operation illustrating combination of a motif/basis with a point lattice to generate a crystal structure.

Bloch Sphere

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Bose Einstein Distribution

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Bose Einstein Distribution 3d

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Branch and Bound

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Illustration of a linear-programming (LP)-based branch and bound algorithm. Not knowing how to solve a mixed-integer programming (MIP) problem directly, it first removes all integrality constraints. This results in a solvable LP called the linear-programming relaxation of the original MIP. The algorithm then picks some variable x restricted to be integer, but whose value in the LP relaxation is fractional. Suppose its value in the LP relaxation is x = 0.7. It then excludes this value by imposing the restrictions x ≤ 0 and x ≥ 1, thereby creating two new MIPs. By applying this recursively step and exploring each resulting bifurcation, the globally optimal solution satisfying all constraints can be found.

Branch Cuts 1

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Branch Cuts 2

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Branch cuts approaching the real $q_0$-axis with decreasing $k$ where they merge at $k = 0$. $k$ denotes the energy cutoff of a scalar field theory.

Change of Variables

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Closed String Topologies

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When calculating scattering amplitudes via the path integral, we must sum over all possible world-sheet topologies. To characterize the types of world-sheets that have to be considered at each level of its perturbative expansion, string theory makes use of the following theorem:

Every compact, connected, oriented two-dimensional manifold is topologically equivalent to a sphere with $g$ handles ($g$ for genus) and $b$ boundaries. A topological invariant of two-dimensional oriented surfaces is the Euler characteristic $\chi = 2 - 2g - b$.

Complex Sign Function

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3d surface plot of the complex sign function $s(p_0) = \sign(\Re p_0 \, \Im p_0)$ over the complex plane. Used in the Matsubara summation of thermal quantum field theory to split contour integrals in the complex plane into two parts, the first being branch-cut free and the second evident branch cut structure.

Complex Sign Plane

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Sign function in the complex plane.

Concave Functions

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$x \ln(x)$ is convex and $x - 1$ is its tangent, so $x \ln(x) \geq x - 1 \enskip \forall x$.

Convex Functions

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Convex Hull of Stability

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Reproduced from Bartel et al. http://dx.doi.org/10.1007/s10853-022-06915-4. The convex hull of stability represents the lowest energy surface in composition space for a given chemical system. It is constructed by connecting the energies of the most thermodynamically stable compounds at each composition. For visual clarity, only ground-state phases at each composition are shown. Legend:

• Blue circles: Stable phases lying on the convex hull (blue line)
• Red squares: Unstable phases above the hull
• Orange dashed line: Chemical potential range where AX is stable
• ΔE_d: Decomposition enthalpy, the energy above the hull for unstable phases

Materials on the hull are stable against decomposition into other phases. The energy difference between a material and the hull (hull distance) quantifies its thermodynamic stability - larger distances meaning less stable and harder to synthesize, making the convex hull a powerful tool for guiding experimental synthesis efforts.

Cost vs Accuracy in Quantum Mechanical Simulations

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Critical Temperature

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A plot illustrating the temperature-dependent phase transitions of a material as a function of the critical temperature ($T_c$). The blue curve represents the low-temperature phase, the red curve represents the high-temperature phase, and the orange curve shows the critical mass ($m_c$) as a function of temperature. This visualization helps to understand the behavior of materials as they undergo phase transitions due to changes in temperature.

Cylinder to Plane

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String theory: Primary fields and radial quantization A time-ordered product of fields on the cylinder maps to a radially ordered product in the complex plane. This graphic visualizes how different times on the cylinder correspond to different times on the plane.

DFT Choices

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A diagram illustrating the various choices and components in density functional theory (DFT) calculations, including kinetic energy, external potential, Hartree potential, and exchange-correlation terms.

DFT Jacob's Ladder

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A visualization of Perdew's Jacob's ladder of density functional theory (DFT), representing a hierarchy of increasingly sophisticated approximations to the exchange-correlation energy (E_xc). The ladder spans from simple local density approximations at the bottom to accurate but expensive double-hybrid functionals near the top, with each rung adding new physical ingredients to improve accuracy at the cost of computational complexity.

DFT MLFF CFF Speed Accuracy Transfer

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This spider diagram compares three computational chemistry methods: Classical Force Fields, Foundational ML Force Fields, and Density Functional Theory (DFT). The comparison is based on three key attributes:

1. Accuracy: The precision and reliability of the method's predictions.
2. Speed: The computational efficiency and time required for simulations.
3. Generalizability: The ability to be applied across diverse chemical systems and material classes.

• Classical Force Fields (red): Highest speed, medium accuracy, lowest generalizability.
• DFT (green): Highest accuracy and generalizability, but lowest speed.
• Foundational ML Force Fields (blue): Balanced performance across all attributes, positioned between classical methods and DFT.

This diagram shows common trade-offs, highlighting how Foundational ML Force Fields aim to bridge the gap between the speed of classical methods and the accuracy of DFT, while offering improved generalizability over classical force fields.

Diagrams

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Disk to Plane

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Divergence

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Plot of $-\ln(1-z)$ which diverges at $z = 1$.

Dropout

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Illustration of applying dropout with a rate of $p = 0.5$ to a multilayer perceptron.

Ergodic

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Euler Angles

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Named after the mathematician Leonhard Euler, the 3 Euler angles describe arbitrary orientations in 3d w.r.t. to some coordinate system. They can also represent the orientation of a frame of reference in physics or of a general basis in 3d linear algebra.

Fermi-Dirac Distribution

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A plot illustrating the Fermi-Dirac distribution function for different temperatures relative to the chemical potential ($\epsilon/\mu$). The plot demonstrates how the occupation number ($n(\epsilon)$) changes as the temperature increases from zero to higher values. The visualization also highlights the effect of thermal fluctuations, which increase with higher temperatures.

Ferroelectric Response

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Feynman Diagram 1

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Feynman Diagram 2

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Feynman Diagram 3

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Feynman Diagram 4

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Feynman Diagram Loop

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Feynman Diagram Loops

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Feynman Diagram One Point

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This Feynman diagram corresponds to the integrand in this expression for $\displaystyle \partial_t \Gamma_{k,a}^{(1)}(q) = -\frac{1}{2} \sum_{\substack{i,j\\k,l}}^N \int_{\substack{p_1,p_2\\p_3^\prime,p_4^\prime}} \frac{\partial_t R_{k,ij}(p_1,p_2)}{\Gamma_{k,jk}^{(2)}(p_2,p_3) + R_{k,jk}(p_2,p_3)} \, \frac{\Gamma_{k,akl}^{(3)}(q,p_3,p_4)}{\Gamma_{k,li}^{(2)}(p_4,p_1) + R_{k,li}(p_4,p_1)}.$

Feynman Diagram Propagator Loop

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This Feynman diagram contains two propagators forming a loop carrying the external energy $q_0$. $m_{1,2}$ denote the masses of the propagators and $\gamma_{1,2}$ their decay width which, for an expansion in Minkowski space, are non-zero only around a real and positive on-shell frequency $p_0 > 0$.

Four Vs of Data

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Gas Pressure on Wall

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Exercise illustration: Compute the pressure of an ideal gas in three dimensions upon a wall at $x = 0$ that attracts molecules at large distance and repels them at smaller distance. Let the force be given by the potential $U(x) = -A \, e^{-\alpha x} + B \, e^{-2 \alpha x}$, with $A,B > 0$.

Generative Adversarial Network

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Generative adversarial network (GAN) architecture. A GAN has two parts. The discriminator $D$ acts as a classifier that learns to distinguish fake data produced by the generator $G$ from real data. $G$ incurs a penalty when $D$ detects implausible results. This signal is backpropagated through the generator weights such that $G$ learns to produce more realistic samples over time, eventually fooling the discriminator if training succeeds.

Geometric Bayes

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GNN Node Aggregation

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Diagram illustrating the message-passing process in graph neural networks (GNNs). Shows how information flows from neighboring nodes to a target node A through multiple layers of aggregation. The first layer combines information from direct neighbors (B, C, D) while the second layer incorporates information from nodes two hops away (E, F), demonstrating how a node's receptive field grows with network depth.

Grand Canonical Occupation Fluctuations

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Graph Isomorphism

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Graphs can look differently but be identical. The only thing that matters are which nodes are connected to which other nodes. If there exists an edge-preserving bijection between two graphs, in other words if there exists a function that maps nodes from one graph onto those of another such that the set of connections for each node remain identical, the two graph are said to be isomorphic.

Gravitons

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Harmonic Oscillator Energy vs Angular Frequency

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Harmonic oscillator energy $\langle E\rangle/k_\text{B} T$ vs angular frequency $\omega = \sqrt{k/m}$

Harmonic Oscillator Energy vs inverse Temperature

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Harmonic oscillator energy vs inverse temperature beta = 1/(kB T)

Heatmap

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Higgs Potential

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High Entropy Alloy

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Isotherms

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This image visualizes the behavior of gas isotherms (curves showing pressure-volume relationships at constant temperature), represented by three equations of state, as a function of volume. The equations include the ideal gas law, and two other modified laws accounting for molecular interactions. The x-axis represents the molar volume and the y-axis represents the pressure of the gas.

Jensens Inequality

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This graph illustrates Jensen's Inequality, a fundamental concept in mathematical optimization and probability theory. It compares a convex function (blue curve, representing the natural logarithm function) with a linear function (orange dashed line). The inequality states that for a convex function, the function of an expectation is always less than or equal to the expectation of the function, depicted here by the fact that the dashed line is always above the curve. Equality holds if and only if the random variable is a constant (i.e. there is no randomness).

k-Space

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The $k$-space region of wave vectors up to a given magnitude is a disk of area $\pi k^2$. The $k$-space area occupied by a single particle state is

where the factor of $\frac{1}{2}$ is due to the spin degeneracy $g_s = 2 s + 1 = 2$. The number of states with wave vector magnitude smaller than $k$ is

$k$-space is used extensively in solid state physics e.g. to visualize the energy bands of a material as a function of electron momentum. In position space, the position-energy dispersion would just be a probability blur.

Kohn Sham Cycle

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This diagram illustrates the Kohn-Sham cycle in Density Functional Theory (DFT), a computational quantum mechanical modeling method. The process starts with an initial guess for the electron density, which is used to calculate the effective potential. This potential is then used in the Kohn-Sham Hamiltonian, which is solved to obtain the wavefunctions. These wavefunctions are used to compute a new electron density. If convergence criteria are met, the final electron density is used to calculate the total energy functional. If not, the new electron density is used for the next iteration of the cycle.

Light

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Visualization of a linearly polarized electromagnetic wave propagating in the z-direction, showing the orthogonal electric (E) and magnetic (B) field components oscillating in phase. The electric field (blue) and magnetic field (red) are perpendicular to each other and to the direction of propagation, forming a transverse wave as described by Maxwell's equations. This is a fundamental representation of electromagnetic radiation, which includes visible light, radio waves, X-rays, and other forms of electromagnetic waves that travel at the speed of light in vacuum.

M-Theory

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MADE

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MAF

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Materials Informatics

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Materials Informatics Challenges

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Matsubara Contour 1

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Complex contour plot illustrating a Matsubara summation. Used in thermal quantum field theory to compute Feynman diagrams at non-zero temperature. $C$ surrounds the imaginary $p_0$-axis counterclockwise but excludes poles of $(-p_0^2 + x^2)^{-1}$.

Matsubara Contour 2

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Matsubara Contour 3

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Matsubara Contour 4

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Matsubara Contour 5

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Matsubara Contour Deformation

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Maxwell Boltzmann Distribution

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The Maxwell-Boltzmann distribution plotted at different temperatures reveals that the most probable velocity $v_p$ of ideal gas particles scales with the square root of temperature.

Mexican Hat

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ML Activations

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Plot of common machine learning activation functions. Includes the Rectified Linear Unit (ReLU), Gaussian Error Linear Unit (GELU), Leaky ReLU, Sigmoid, and Hyperbolic Tangent (tanh) activation functions. Activation functions introduce non-linearity into neural networks, enabling them to learn complex patterns and relationships in data. The choice of activation function can have a significant impact on the performance and convergence of a neural network. ReLU is widely used due to its simplicity and ability to alleviate the vanishing gradient problem. GELU and Leaky ReLU are variants of ReLU that address some of its limitations. Sigmoid and tanh are classical activation functions that squash the input to a fixed range, but they are less commonly used in modern deep learning architectures due to the vanishing gradient problem.

Momentum Shell

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See also https://diagrams.janosh.dev/ergodic.

MOSFET

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The metal-oxide-semiconductor field-effect transistor, or MOSFET for short, is the most frequently manufactured device in history, with close to 10^23 MOSFETs produced since 1960. As the basic building block of almost all modern electronics, it revolutionized the world economy and triggered our ascent into the information age.

Normalizing Flow

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Normalizing Flow Coupling Layer

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Open String Topologies

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When calculating scattering amplitudes via the path integral, we must sum over all possible world-sheet topologies. To characterize the types of world-sheets that have to be considered at each level of its perturbative expansion, string theory makes use of the following theorem:

Every compact, connected, oriented two-dimensional manifold is topologically equivalent to a sphere with $g$ handles ($g$ for genus) and $b$ boundaries. A topological invariant of two-dimensional oriented surfaces is the Euler characteristic $\chi = 2 - 2g - b$.

Operator Orderings

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Organic Molecule

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Otto Cycle

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The Otto cycle is an idealized thermodynamic cycle of a typical spark ignition piston engine. It is the thermodynamic cycle most commonly found in automobile engines.

Periodic Table

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The periodic table of elements, a graphic formulation of the periodic dependence of elemental properties on their atomic numbers. The table is divided into four roughly rectangular areas called blocks. Rows are called periods, columns are groups. Elements from the same group show similar chemical characteristics. Trends run through the periodic table, with nonmetallic character (keeping their own electrons) increasing from left to right across a period, and from down to up across a group, and metallic character (surrendering electrons to other atoms) increasing in the opposite direction.

Physics Mindmap

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A mindmap showing the main branches of physics, including Classical Mechanics, Quantum Mechanics, Relativity, Statistical Mechanics, High-Energy Physics, and Cosmology. Each branch is further divided into subcategories.

Plane to Torus

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Plate Capacitor

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Parallel plate capacitor with dipolar polarization.

Potential Triangle

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Propagator Fluctuations

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QFT Propagator Poles

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QFT Propagators

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Random Forest

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Diagram of the random forest (RF) algorithm (Breiman 2001). RFs are ensembles model consisting of binary decision trees that predicts the mode of individual tree predictions in classification or the mean in regression. Every node in a decision tree is a condition on a single feature, chosen to split the dataset into two so that similar samples end up in the same set. RFs are inspectable, invariant to scaling and other feature transformations, robust to inclusion of irrelevant features and can estimate feature importance via mean decrease in impurity (MDI).

Regular vs Bayes NN

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RG Flow in Theory Space

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Visualization of renormalization group (RG) flow trajectories in the theory space of all possible coupling constants (λ₁, λ₂, λ₃, λ₄) allowed by symmetries. The diagram shows how different microscopic actions (S₁, S₂, S₃) flow to the same quantum effective action (Γ) through different paths, each corresponding to a different choice of infrared regulator R_k(q). This illustrates the universality of the effective action despite different starting points and regulators.

RNVP

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Roost Update

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Sabatier Principle

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Saddle Point

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This graph depicts a saddle point in three-dimensional space, which is a point in the domain of a function that is a local minimum in one direction and a local maximum in another. The depicted function $F(T, V)$ is quadratic in $T$ and $V$, and the graph shows the convex and concave nature of the function along different axes.

Scattering Detailed Balance

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A simple diagram representing a scattering process in particle physics, where two incoming particles with momenta $p$ and $k$ interact and produce two outgoing particles with momenta $p'$ and $k'$. The principle of detailed balance states that at equilibrium, each scattering process is in equilibrium with its reverse process.

Seebeck Effect

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The Seebeck effect creates a voltage in conductors or semiconductor when subjected to a temperature gradient, i.e. when one side of the material is warmer than another. This effect enables thermoelectric devices which can measure temperature and generate electricity from waste heat.

Self Attention

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Single-head attention

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Flow diagram of single-head attention illustrating the equation $\displaystyle \mathrm{Attention}(Q, K, V) = \mathrm{softmax}_\text{row} \left( \frac{Q K^\top}{\sqrt{d}} \right) V$ with border colors to indicate tensor dimensions.

Skip Connection

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Spontaneous Magnetization

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Spontaneous magnetization $m$ in an Ising ferromagnet appears below the critical temperature $T_c$. The derivative of $m$ diverges in the limit $T \to T_c^-$.

Statistical Energy Distributions

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A plot comparing the identical particle distribution functions of Bose-Einstein, Boltzmann, and Fermi-Dirac statistics as a function of the reduced chemical potential $\beta (\epsilon - \mu)$. This visualization highlights the differences between the three types of distribution functions, which are used to describe the behavior of particles in different statistical systems.

Tanh

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Plot of the hyperbolic tangent. tanh(x) goes to +/- 1 for x to +- infinity while it is approximately linear for abs(x) < 1. Appears in many places in physics, e.g. in the magnetization of an ideal paramagnet of independent spins.

Thermodynamic Ensemble transforms

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Equivalence of thermodynamic ensembles through Laplace and Legendre transforms.

Thermodynamic Potentials Venn Diagram

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A Venn diagram illustrating the relationships between thermodynamic potentials, forces, and flows. The diagram shows three main regions (Mechanical, Thermal, Chemical) and their intersections, representing different thermodynamic potentials. The subscript μ denotes an N-to-μ Legendre transform of the indicated potential (U, H, F, or G). The diagram helps visualize how different thermodynamic quantities are related through Legendre transforms.

Thomson Scattering

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Illustration of coherent X-ray scattering (i.e. Thomson scattering) from an atom. An electron is caused to oscillate and then re-emits a photon with the same energy in an arbitrary direction.

Tori

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Torus

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Torus Fundamental Domain

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Two Point

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Two-point propagator flow

Two Point No Cutoff

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Two-point propagator flow without cutoff derivative

Unregularized Propagator Diagrams

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Variational Autoencoder

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Wetterich Equation

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The Wetterich eqn. is a non-linear functional integro-differential equation of one-loop structure that determines the scale-dependence of the flowing action $\Gamma_k$ in terms of fluctuations of the fully-dressed regularized propagator $[\Gamma_k^{(2)} + R_k]^{-1}$. It admits a simple diagrammatic representation as a one-loop equation as shown in this diagram.

Wyckoff Positions

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zT vs n

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