Single Walker Observables and Probability Fields#
0. TLDR#
Single Walker Observable Stack: We formalize the observable map of a single walker (Definition 428) inside a swarm (Definition 429). The stack couples positional, velocity, and potential data through a fixed parameter set \(\Theta_{\text{obs}}\) and produces the diversity, reward, and fitness channels used by the Fragile Gas.
Diversity Channel Distribution: The distribution \(\mathcal{P}_{D(i)}\) of the diversity channel is an explicit sum over all companion maps produced by the sequential greedy pairing routine (including the self-pair when \(k\) is odd or \(k=1\)). Each term factors into the probability of the pairing realization and a deterministic evaluation of the regularized distance statistic.
Conditional Cloning Probability Field: For every potential companion \(j\), the cloning pipeline produces a random variable \(\Pi(i|j)\) whose randomness comes from diversity pairing realizations; the cloning companion draw itself is independent. The family \(\{\mathcal{P}_{\Pi(i|j)}\}_{j}\) is the discrete probability field controlling local exploration pressure (Definition 539).
Post-Cloning Mixture: The position of walker \(i\) after the cloning operator (Definition 546) follows a mixed distribution with a Dirac mass at the original position and a Gaussian mixture centered at the companions. The weights are the joint probabilities of selecting each companion and executing a cloning action.
Kinetic Coupling: Convolving the post-cloning mixture with the BAOAB kinetic propagator (Definition 583) produces the full single-step transition kernel consistent with the kinetic operator (Definition 558). This kernel directly yields the single walker death probabilities through boundary integrals over \(\mathcal{X}_{\mathrm{valid}}\) (Definition 431).
Dependencies: Mathematical Foundations of Fragile
1. Introduction#
1.1. Goal and Scope#
This chapter isolates the single walker perspective inside the Euclidean Gas. Starting from a fixed swarm configuration \(S_t = ((x_i, v_i, s_i))_{i=1}^N\), we describe every random object that influences the evolution of a specific alive walker \(i \in \mathcal{A}(S_t)\) (Definition 430). The objective is to turn the algorithmic pipeline—diversity measurement, reward assessment, cloning, and kinetic motion—into explicit probability distributions on \(\mathbb{R}^d\) that can be used for both analysis and simulation.
1.2. Relationship to the Framework#
The macroscopic drift and Lyapunov statements of Chapter 3 rely on aggregate statistics such as the positional variance or the hypocoercive Wasserstein distance. The present chapter supplies the microscopic objects that feed those statistics:
The diversity channel captures local geometric information via the algorithmic distance (Definition 448).
The reward channel translates the energy landscape and velocity regularization into standardized scores.
The fitness potential combines both channels multiplicatively to determine competitive outcomes.
The conditional cloning field and post-cloning mixture describe the actual transitions used in Definition 546.
The kinetic convolution produces the law of the full update operator (Definition 478) at the single walker level.
All expressions below are consistent with the Fragile Gas axioms (Definition 483) and adopt the same notation as Chapter 3.
1.3. Notation#
\(\mathcal{A}(S_t)\) denotes the set of alive walkers at time \(t\) with cardinality \(k = |\mathcal{A}(S_t)|\).
\(c_j\) is the companion of walker \(j\) in the diversity companion map \(M \in \mathcal{M}(\mathcal{A}(S_t))\); if \(k\) is odd (or \(k=1\)), the leftover walker is mapped to itself, \(c_j=j\).
\(C_i^{\text{div}}\) is the random choice used by the diversity pairing routine.
\(C_i\) is the cloning companion for walker \(i\), drawn independently of \(M\) from a separate kernel that excludes \(i\) whenever possible (so \(P(C_i=i)=0\) if \(k>1\)).
\(d_{\text{alg}}\) is the algorithmic distance with weight \(\lambda_{\text{alg}}\) (Definition 527).
\(Z(\cdot)\) and \(Z_r(\cdot)\) denote standardized statistics with a patched variance \(\sigma'\), ensuring the denominators are strictly positive.
Bold symbols represent deterministic functions; calligraphic symbols denote distributions or fields.
2. The Single Walker Parameter Stack#
We fix the parameter set
The first block controls the diversity and reward channels, the second block fixes cloning companion selection, collision, and thresholding, and the last block governs the kinetic step. The diversity pairing and cloning companion selection are treated as independent draws (typically with \(\epsilon_c\) tuned separately from \(\epsilon_d\)). Additional algorithm-specific constants are absorbed into \(\Theta_{\text{obs}}\) but are omitted from the notation when not required.
Definition 493 (Observable Parameter Stack)
Given a swarm state \(S_t\), the observable stack of walker \(i\) is the tuple
where each component is defined in Sections 3–7 below. Every map depends measurably on \(S_t\) and on the algorithmic randomness of the diversity pairing, cloning companion selection, cloning threshold, collision rotations, and the kinetic noise.
3. Distribution of the Diversity Channel#
3.1. Matching Probabilities#
The sequential greedy pairing algorithm produces a companion map \(M\) of \(\mathcal{A}(S_t)\). When \(k\) is even, \(M\) is a perfect matching. When \(k\) is odd, the implementation returns an involution with one fixed point: the leftover walker is mapped to itself, \(c_i=i\), which keeps the distance measurement well-defined because of the \(\epsilon_{\text{dist}}\) regularizer. When walker \(i\) is processed and chooses among the unpaired set \(U \subseteq \mathcal{A}(S_t)\), the selection probability for \(j \in U \setminus \{i\}\) is
If \(U \setminus \{i\} = \varnothing\) (e.g., \(k=1\)), the implementation sets \(c_i=i\).
Let \(\mathcal{I}(M)\) denote the ordered list of walkers processed as the “first” element of each pair. The probability of a pairing realization \(M \in \mathcal{M}(\mathcal{A}(S_t))\) is the product of the sequential choices that realize those pairs, conditioned on the processing order used by the greedy algorithm (the fixed point, if present, contributes no factor):
where \(U_i(M)\) records the remaining unmatched walkers when \(i\) is processed.
Remark 112 (Independent Companion Selection)
If the diversity channel uses an independent softmax companion selector instead of mutual pairing, interpret \(M\) as the vector of companion indices \((c_i)_{i \in \mathcal{A}(S_t)}\). Then
with the same self-pairing fallback when only one alive walker is available. The downstream definitions of \(d_i\), \(Z(i, M, S_t)\), and \(\mathcal{P}_{D(i)}\) are unchanged. The cloning companion draw \(C_i\) remains an independent kernel (Section 5) and is not tied to this diversity map.
3.2. Regularized Distance Measurement#
For every walker \(j\), the regularized distance to its companion is
The regularization prevents degeneracy when walkers coincide and ensures differentiability of the subsequent statistics.
3.3. Standardization Regimes#
Let \(k = |\mathcal{A}(S_t)|\). Two regimes are supported:
Global statistics (\(\rho = \mathrm{None}\)):
\[ \mu_d(M, S_t) = \frac{1}{k} \sum_{j \in \mathcal{A}(S_t)} d_j(M; \cdot), \quad \sigma'_d(M, S_t; \sigma_{\text{min}}) = \sqrt{\frac{1}{k} \sum_{j \in \mathcal{A}(S_t)} (d_j - \mu_d)^2 + \sigma_{\text{min}}^2}.\]Localized statistics (\(\rho < \infty\)):
\[ K_{\rho}(i, j) = \exp\left(-\frac{d_{\text{alg}}(i, j; \lambda_{\text{alg}})^2}{2\rho^2}\right), \quad \mu_{\rho, d}(i) = \frac{\sum_{j \in \mathcal{A}(S_t)} K_{\rho}(i, j) d_j}{\sum_{\ell \in \mathcal{A}(S_t)} K_{\rho}(i, \ell)},\]\[ \sigma'_{\rho, d}(i; \sigma_{\text{min}}) = \sqrt{\frac{\sum_{j \in \mathcal{A}(S_t)} K_{\rho}(i, j) (d_j - \mu_{\rho, d}(i))^2}{\sum_{\ell \in \mathcal{A}(S_t)} K_{\rho}(i, \ell)} + \sigma_{\text{min}}^2}.\]
3.4. Diversity Distribution#
The standardized score of walker \(i\) is
After applying the logistic rescale and positivity floor, the diversity channel is
The probability mass function of \(\mathcal{D}_i\) is therefore
Each atom corresponds to a companion map realization whose deterministic evaluation equals \(v\).
4. Reward Channel and Fitness Potential#
4.1. Raw Reward#
The deterministic reward of walker \(j\) uses the positional reward potential \(R_{\text{pos}}\) and the velocity regularization coefficient \(c_{v\_\text{reg}}\):
The kinetic potential \(U\) used by the Langevin operator may be different from \(R_{\text{pos}}\); the reward channel does not assume a shared potential.
4.2. Reward Standardization#
The global statistics follow the same form as Section 3.3, using the full alive set. For a localized regime, the same kernel \(K_{\rho}\) is reused:
The standardized score \(Z_r(i, S_t)\) follows from the corresponding means and variances.
4.3. Reward Channel Value#
The reward channel is deterministic once \(S_t\) is fixed:
4.4. Fitness Potential Distribution#
Proposition 121 (Single Walker Fitness Distribution)
The fitness potential \(V_{\text{fit}}(i, M, S_t) = \mathcal{D}_i(M, S_t) \cdot R_i(S_t)\) has probability mass function
Therefore, \(\mathcal{P}_{V(i)}\) inherits the atomic structure of \(\mathcal{P}_{D(i)}\) scaled by the deterministic reward factor.
5. Conditional Cloning Probability Field#
5.1. Cloning Scores#
Given a diversity pairing realization \(M\), the cloning score of walker \(i\) relative to a cloning companion \(j\) is (Definition 538)
This value becomes a random variable once \(M\) is sampled.
5.2. Thresholding and Probabilities#
Let \(T_i \sim \mathrm{Uniform}(0, p_{\max})\) be the stochastic threshold. The clipped probability (Definition 539) is
Conditioned on \(M\) and a companion \(j\), the cloning action indicator is \(\mathbf{1}_{\{ s(i \mid j, M) > T_i \}}\) (Definition 540), and \(\mathbb{P}(\text{clone} \mid s(i \mid j, M)) = \pi_{\text{clip}}(s(i \mid j, M); p_{\max})\).
5.3. Field Definition#
The conditional cloning probability field of walker \(i\) is the map
where
Its expectation
determines the mean contribution of \(j\) to the total cloning probability \(\pi_{\text{clone}}(i \mid S_t)\). The cloning companion draw \(C_i\) is independent of the diversity pairing \(M\) and excludes \(i\) whenever possible, so \(P(C_i=i)=0\) if \(k>1\). The total probability is
consistent with Definition 539. Here \(P_{C_i}\) is the cloning companion selection distribution (Definition 537).
Conditional copy probabilities and realized events
Rust engine experiment · Seed 7 · Poster after 7 experiment steps. Interactive view starts from the same seed.
6. Post-Cloning Position Distribution#
6.1. Mixture Structure#
Let \(x_i\) denote the original position of walker \(i\). After the cloning operator, the position \(X'_i\) has the mixed distribution
where \(\delta\) is the Dirac mass and \(\mathcal{N}(\cdot; x_j, \sigma_x^2 I_d)\) is the isotropic Gaussian induced by the spatial jitter.
When \(k>1\), \(P(C_i=i)=0\) so the self-centered Gaussian component drops out; when \(k=1\), the only potential cloning companion is \(i\) itself and its weight is determined by \(\bar{p}(i \mid i)\).
6.2. Joint Weights#
The joint weights factorize as
and obey \(\sum_{j} w_{ij}^{\text{joint}} = \pi_{\text{clone}}(i \mid S_t)\). Each Gaussian component is therefore weighted by the probability of selecting \(j\) as a companion and subsequently cloning against \(j\).
6.3. Post-Cloning Velocity Distribution#
Let \(C_i=j\) and let \(I_c\) denote the collision group associated with companion \(c=j\) (so \(i \in I_c\)). The inelastic collision update (Definition 541) gives
where \(V_{\text{COM},c} = \frac{1}{|I_c|+1}\left(v_c + \sum_{\ell \in I_c} v_\ell\right)\) and \(R_i^{\text{rot}}\) is a random rotation (uniform on \(SO(d)\)). Conditional on \(S_t\) and the collision group \(I_c\), \(V'_i\) is supported on a sphere of radius \(\alpha_{\text{restitution}}\|u_i\|\) centered at \(V_{\text{COM},c}\). The positional jitter and the rotation draws are independent, so the conditional joint law \(\mathcal{P}_{X'_i, V'_i}\) factors into the Gaussian position kernel from Section 6.1 and the spherical velocity law above. When \(k=1\), the collision group is trivial and \(V'_i = v_i\).
Remark 113 (Interpretation as a Discrete Random Field)
The function \(x' \mapsto \mathcal{P}_{X'_i}(x')\) is a random field centered on the alive swarm: its Gaussian components are centered at alive positions but the distribution has full support on \(\mathbb{R}^d\). Locations near highly informative companions inherit large Gaussian weights, while inactive regions contribute only through the persistence mass. This view is convenient when studying local exploration pressure or extinction risk inside restricted domains.
Track the actual cloning displacement
Rust engine experiment · Seed 7 · Poster after 7 experiment steps. Interactive view starts from the same seed.
7. Coupling with the Kinetic Operator#
7.1. BAOAB Convolution#
Let \((X'_i, V'_i)\) denote the state immediately after cloning. The kinetic operator \(\Psi_{\text{kin}}\) in Stratonovich form (Definition 558) is implemented numerically through the BAOAB update (Definition 583). Let \(\mathcal{K}_{\text{BAOAB}}^x(x'' \mid x', v')\) denote the positional marginal of the BAOAB kernel. In the reference implementation, this kernel is the pushforward of Gaussian noise through the full BAOAB map (including potential forces and any optional adaptive terms), so it need not be Gaussian when the force field is nonlinear. The full single-step transition law is the convolution
where the joint distribution \(\mathcal{P}_{X'_i, V'_i}\) is induced by the cloning operator (positional marginal in Section 6.1, velocity update in Section 6.3).
7.2. Death Probability Field#
Let \(\mathcal{X}_{\mathrm{valid}} \subset \mathbb{R}^d\) be the viable domain (Definition 431). For each companion \(j\) let \(E_{ij}\) denote the event that \(C_i=j\) and the cloning action indicator is 1. The conditional probability of exiting \(\mathcal{X}_{\mathrm{valid}}\) after cloning to \(j\) and applying the kinetic step is
where \(\mathcal{P}_{X''_i \mid E_{ij}, S_t}\) is obtained by convolving the conditional post-cloning law of \((X'_i, V'_i)\) given \(E_{ij}\) with the positional BAOAB kernel \(\mathcal{K}_{\text{BAOAB}}^x\). The total death probability is the weighted sum
with the persistence term computed by integrating the kinetic kernel based on the Dirac mass at \((x_i, v_i)\).
8. Parameter Glossary#
Parameter |
Symbol |
Unit |
Role |
Primary Reference |
|---|---|---|---|---|
Algorithmic distance weight |
\(\lambda_{\text{alg}}\) |
\([\text{dimensionless}]\) |
Balances position and velocity terms in \(d_{\text{alg}}\) |
|
Diversity interaction radius |
\(\epsilon_d\) |
\([\text{distance}]\) |
Controls softness of the pairing kernel |
Section 3.1 |
Cloning interaction radius |
\(\epsilon_c\) |
\([\text{distance}]\) |
Controls softness of the cloning companion kernel |
|
Distance regularizer |
\(\epsilon_{\text{dist}}\) |
\([\text{distance}]\) |
Prevents degeneracy of \(d_j\) |
Section 3.2 |
Localization scale |
\(\rho\) |
\([\text{distance}]\) |
Chooses between global and local statistics |
Section 3.3 |
Variance patch |
\(\sigma_{\text{min}}\) |
\([\text{dimensionless}]\) |
Ensures positive denominators in Z-scores |
Sections 3.3–4.2 |
Logistic bound |
\(A\) |
\([\text{dimensionless}]\) |
Caps the rescaled channel output |
Sections 3.4–4.3 |
Positivity floor |
\(\eta\) |
\([\text{dimensionless}]\) |
Guarantees strictly positive channels |
Sections 3.4–4.3 |
Reward exponent |
\(\alpha\) |
\([\text{dimensionless}]\) |
Shapes the non-linearity of the reward channel |
Section 4.3 |
Diversity exponent |
\(\beta\) |
\([\text{dimensionless}]\) |
Controls sensitivity of the diversity channel |
Section 3.4 |
Reward velocity penalty |
\(c_{v\_\text{reg}}\) |
\([\text{dimensionless}]\) |
Penalizes high kinetic energy in \(r_j\) |
Section 4.1 |
Reward potential |
\(R_{\text{pos}}(\cdot)\) |
\([\text{dimensionless}]\) |
Positional reward used in \(r_j\) |
Section 4.1 |
Kinetic potential |
\(U(\cdot)\) |
\([\text{dimensionless}]\) |
Drives the Langevin dynamics |
Section 7.1 |
Threshold scale |
\(p_{\max}\) |
\([\text{dimensionless}]\) |
Upper bound of the cloning threshold; sets clipping scale |
Section 5.2 |
Cloning denominator regularizer |
\(\epsilon_{\text{clone}}\) |
\([\text{dimensionless}]\) |
Prevents division by zero in \(s(i \mid j, M)\) |
Section 5.1 |
Spatial jitter |
\(\sigma_x\) |
\([\text{distance}]\) |
Standard deviation of positional perturbations during cloning |
Section 6.1 |
Restitution coefficient |
\(\alpha_{\text{restitution}}\) |
\([\text{dimensionless}]\) |
Inelastic collision contraction factor |
Section 6.3 |
Kinetic friction |
\(\gamma\) |
\([1/\text{time}]\) |
BAOAB damping coefficient |
Section 7.1 |
Kinetic inverse temperature |
\(\beta_{\text{kin}}\) |
\([\text{dimensionless}]\) |
Scales the kinetic noise term |
Section 7.1 |
Time step |
\(\Delta t\) |
\([\text{time}]\) |
Temporal discretization of the kinetic operator |
Section 7.1 |
The table collects the parameters that appear explicitly in the observable stack, enabling traceability between algorithmic implementation and the formal probability objects derived above.