Thermoparticles: Dynamics in Thermal Gradients

• Thermoparticles are particle-like excitations whose behavior is fundamentally altered by temperature gradients and non-equilibrium effects across soft matter and quantum systems.
• They exhibit measurable phenomena such as thermophoretic drift in colloids, underdamped oscillations in mesoscopic systems, and medium-modified resonances in finite-temperature quantum field theory.
• Their study employs techniques from optical trapping to lattice QCD, integrating non-equilibrium statistical mechanics and stochastic thermodynamics to reveal insights into particle transport and phase transitions.

A thermoparticle is a particle-like excitation, object, or colloidal entity whose dynamics, transport, or spectral properties are fundamentally altered by the presence of a thermal gradient or thermal medium. The term is used in multiple contexts—ranging from soft condensed matter physics, where it denotes colloids or nanoparticles subjected to temperature gradients and exhibiting thermophoretic motion, to quantum field theory, where it characterizes medium-modified single-particle (or collective) excitations at finite temperature that persist as resonances or broadened poles in spectral functions. Thermoparticles are distinguished from conventional particles or quasiparticles by the essential role of temperature, non-equilibrium statistical mechanics, and dissipation or screening in their existence and properties.

1. Thermoparticles in Soft Matter: Colloids, Nanoparticles, and Thermophoresis

In soft condensed matter, thermoparticles concretely refer to colloidal particles (spheres, Janus particles, nanoparticles) in a fluid subject to static or controlled temperature gradients. Thermophoresis—the resultant drift of such particles along ∇T—is governed by the Soret coefficient $S_T = D_T/D$, defined as the ratio between the thermal diffusion coefficient $D_T$ and the ordinary Brownian diffusion coefficient $D$ (Helden et al., 2014, Huang et al., 2015, Liang et al., 2021). A key phenomenological relation is

$J = -D \nabla n - D_T n \nabla T,$

with $J$ the particle flux and $n$ the local density (Huang et al., 2015, Helden et al., 2014).

At the microscopic level, interfacial effects (surface tension, solvation forces) in the boundary layer around the particle drive a slip flow, imparting a constant thermophoretic drift velocity $v_{\rm th} = -D_T \nabla T$ (Helden et al., 2014). These effects can be measured at the single-particle level using evanescent-light scattering (TIRM), enabling extraction of $S_T$ and the local thermophoretic force with fN precision (Helden et al., 2014).

The sign and magnitude of $S_T$—thermophobic ($S_T>0$, particles move to cold) or thermophilic ($S_T<0$, particles move to hot)—depend sensitively on the interplay between interfacial energetics, particle–solvent interactions, and, for charged systems, the entropy of transfer $\widehat S$ (Huang et al., 2015). In charged colloids, the Soret and Seebeck effects are unified: temperature gradients induce not only particle drift but also large internal electric fields and measurable voltages via the electrolyte Seebeck effect (Huang et al., 2015, Majee et al., 2014).

The response of thermoparticles to external fields or imposed gradients is further complicated in the presence of electrostatic interactions, hydrodynamic confinement, or non-equilibrium state cycling. For instance, under laser heating, non-ionic colloidal spheres develop a net thermocharge $Q$,

$$Q = -e(\alpha_+ - \alpha_-)\frac{a}{\ell_B}\frac{\Delta T}{T},$$

and a long-ranged thermoelectric field, realizing laser-controlled pairwise repulsion, electrophoresis, or selective nanomaterial sorting (Majee et al., 2014).

2. Stochastic Thermodynamics: Thermoparticles as Mesoscopic Systems

Levitated nanoparticles in dilute gas provide a platform where a single thermoparticle (defined as a driven, underdamped Brownian oscillator) becomes the minimal working substance for stochastic thermodynamics (Gieseler et al., 2018, Millen et al., 2020). The dynamics of the center of mass obeys the underdamped Langevin equation,

$$m \ddot x(t) + \gamma \dot x(t) + k x(t) = \sqrt{2\gamma k_B T}\,\xi(t),$$

and enable real-time monitoring of energy, work, and heat at the single-trajectory level (Millen et al., 2020). This realization has supported the experimental verification of fluctuation theorems (e.g. Jarzynski, Crooks), and the construction of microscopic Stirling or Carnot heat engines from single optically trapped beads (Gieseler et al., 2018).

Measurement of work, heat, and entropy production along trajectories allows direct quantification of non-equilibrium thermodynamic quantities. The application of time-dependent traps $k(t)$, feedback cooling, and modulation schemes further probes the response and relaxation properties of mesoscopic thermoparticles, reaching effective temperatures down to $\sim 100\,\mu{\rm K}$ and enabling approaches to the quantum regime (Millen et al., 2020).

3. Thermoparticles in Quantum Field Theory at Finite Temperature

In quantum field theory, thermoparticles are defined through the thermal spectral representation and micro-causality at finite $T$. For a generic mesonic operator $\mathcal O(x)$, the spectral function at temperature $T$ admits a decomposition (Philipsen, 21 Dec 2025, Lowdon et al., 11 Dec 2024, Lowdon et al., 28 Jan 2025):

$$\rho(\omega, \mathbf p) = \int_0^\infty ds \int \frac{d^3u}{(2\pi)^3} \,\varepsilon(\omega)\,\delta(\omega^2 - \mathbf u^2 - s)\,\widetilde D_\beta(\mathbf u, s),$$

with $\widetilde D_\beta$ the thermal spectral density (Lowdon et al., 11 Dec 2024).

When the $T=0$ theory contains a stable particle of mass $m$, at finite $T$ the spectral density splits into a discrete (thermoparticle) contribution and a continuum of multi-particle and collective modes:

$$\widetilde D_\beta(\mathbf u, s) = \widetilde D_{m,\beta}(\mathbf u)\,\delta(s - m^2) + \widetilde D_{c,\beta}(\mathbf u, s)$$

(Philipsen, 21 Dec 2025, Lowdon et al., 11 Dec 2024, Lowdon et al., 28 Jan 2025). The thermoparticle component describes a medium-modified, damped resonance at $s = m^2$ whose amplitude and width encode all multiple-scattering, screening, and collisional effects of the thermal medium.

In hot QCD near and above the chiral crossover $T_{\rm ch}\simeq 156\,{\rm MeV}$, non-perturbative lattice calculations reveal that pseudo-scalar mesons (e.g. $\pi$, $K$) survive as clear resonance-like thermoparticle states for $T_{\rm ch} \lesssim T \lesssim T_{\rm d} \sim 2$–$3\,T_{\rm ch}$ (Philipsen, 21 Dec 2025, Lowdon et al., 11 Dec 2024). Spatial and temporal correlators are quantitatively described by a sum of damped exponential (Yukawa) terms at the vacuum particle masses, with the spectral representation

$$\rho_{\rm TP}(\omega, 0) = \varepsilon(\omega)\,\theta(\omega^2 - m^2)\, \frac{4 \alpha \gamma \sqrt{\omega^2 - m^2}} {(\omega^2 - m^2)^2 + 2\gamma^2 (\omega^2 - m^2) + \gamma^4}$$

(Lowdon et al., 11 Dec 2024, Lowdon et al., 28 Jan 2025). As temperature increases further, these discrete contributions dissolve into a continuum as damping overwhelms resonance formation.

In the context of spontaneously broken symmetries, such as Goldstone bosons in scalar $\phi^4$ theory, the thermoparticle language clarifies that even above the symmetry-restoration temperature, spectral functions retain a non-perturbative, massless, exponentially damped mode—directly observed on the lattice—which is forbidden in simple perturbative treatments (Lowdon et al., 28 Jan 2025).

4. Theoretical Framework: Non-Equilibrium and Thermodynamic Models

The theoretical treatment of thermoparticle dynamics in soft matter and quantum field theory leverages both non-equilibrium statistical mechanics and field theory, with several key results:

• Non-equilibrium nature: Thermophoresis is fundamentally a non-equilibrium phenomenon. In multi-state models, the Soret coefficient can be written as (Liang et al., 2021)

$$S_T = \frac{\partial_T \langle D \rangle_{\rm eq}}{\langle D \rangle_{\rm eq}} = \frac{{\rm Cov}_{\rm eq}(E, D)}{\langle D \rangle_{\rm eq} T^2}$$

with $\langle D \rangle_{\rm eq}$ a Boltzmann average over internal diffusion coefficients $D_i$ and energies $E_i$, and $S_T$ determined by the correlation between energy and mobility.

• Stationary distributions in gradients: The steady-state distribution of a thermoparticle in a temperature gradient generalizes the Boltzmann law, incorporating spatially varying diffusion and external potentials (Maier, 2020):

$$\rho(r) \sim [T(r)]^{-(1-\alpha)} \exp\left[ -\int^{r} \frac{\nabla U}{T(l)} dl \right]$$

where $\alpha$ encodes Ito ($\alpha=0$) vs. Stratonovich ($\alpha=1/2$) stochastic calculus.

• Spectral decomposition: At finite $T$, the spectral function is partitioned into thermoparticle (discrete) and continuum (Landau damping, scattering) pieces, the former dominating at $T \lesssim m_{\rm lightest}$, enabling quantitative extraction of thermal masses, widths, and residues from lattice correlators (Lowdon et al., 11 Dec 2024, Lowdon et al., 28 Jan 2025).
• Thermodynamics in quantum gravity: In loop quantum gravity, thermoparticle properties are computed for both massless and massive particles via analytically tractable modifications of the Bose–Einstein density of states. The resulting thermodynamic functions (pressure, energy, entropy) involve Riemann zeta values and reveal nontrivial Planck-scale dependencies and stability regimes (Filho, 2022).

5. Experimental Techniques and Prototypical Phenomena

Thermoparticles in the laboratory are probed via a variety of experimental techniques:

• Optical trapping and levitation: Nanoparticles are stably levitated and manipulated in well-controlled temperature fields, enabling microgravity-like studies of force balances and rarefied-gas dynamics (Fung et al., 2015, Foster et al., 2017, Millen et al., 2020).
• Evanescent-field scattering (TIRM): Displacement and steady-state distributions are measured for micron-sized beads with fN force sensitivity and sub-micrometer spatial resolution, directly extracting thermophoretic forces and Soret coefficients (Helden et al., 2014).
• Electrolyte Seebeck effect: Local laser heating induces net thermocharge and associated long-range electric fields in colloidal dispersions, quantified using microelectrodes and electrophoretic drift experiments (Majee et al., 2014).
• Lattice QCD correlator analysis: Spectral reconstructions from spatial and temporal lattice correlators enable isolation of thermoparticle contributions, extracting in-medium masses and widths at finite $T$ (Philipsen, 21 Dec 2025, Lowdon et al., 11 Dec 2024).

6. Collective Phenomena, Instabilities, and Applications

Thermoparticle behavior leads to a variety of collective and application-relevant phenomena:

• Pairwise interactions and pattern formation: Laser-heated colloidal thermoparticles interact via long-range $1/r^2$ thermoelectric fields or mutual depletion/accumulation zones, leading to controlled dispersion, aggregation, or colloidal “crystal” arrangements (Majee et al., 2014, Golestanian, 2011).
• Self-propulsion and thermotaxis: Janus colloids produce self-induced temperature gradients, leading to autonomous propulsion (self-thermophoresis) and collective thermotactic instabilities described by nonlinear Poisson–Boltzmann or gravitational-analogue equations (Golestanian, 2011).
• Thermal explosion/instabilities: For negative Soret coefficients ($S_T<0$), systems may exhibit thermophoretic collapse analogously to gravitational systems, with a critical threshold for “thermal explosion,” i.e., run-away clustering and heating (Golestanian, 2011).
• Selective separation: Differential absorption and thermophoretic properties allow for size- and material-selective sorting, for example, the separation of carbon nanotubes by chiral index (Majee et al., 2014).
• Microgravity testbeds: Thermophoretic levitation provides platforms for studying van der Waals, Casimir, and electrostatic forces in a near-force-free environment, with applications to aerosol science and assembly of complex soft-matter structures (Foster et al., 2017, Fung et al., 2015).

7. Implications in Quantum Field Theory and Outlook

Thermoparticles in thermal QFT profoundly impact the understanding of hot, dense matter:

• Redefinition of degrees of freedom: Non-perturbative evidence now indicates that even above the chiral crossover, QCD is not simply a system of deconfined quarks and gluons, but is populated by thermally broadened, damped hadron-like excitations—thermoparticles—whose spectral signatures persist up to $T \sim 2$–$3\,T_{\rm ch}$ (Philipsen, 21 Dec 2025, Lowdon et al., 11 Dec 2024).
• Analytic structure: Unlike the simple poles/breit-wigner forms in perturbation theory, thermoparticles introduce branch point structures in retarded propagators, modifying the large-time and long-distance dynamics of thermal correlators and rendering standard quasiparticle pictures inadequate (Philipsen, 21 Dec 2025).
• Dynamical symmetry emergence: The presence of thermoparticles correlates with emergent chiral-spin $SU(2)_{CS}$ symmetry in QCD, reflecting persistent color-electric confining dynamics and demarcating a window distinct from either the hadron gas or fully deconfined plasma (Philipsen, 21 Dec 2025).
• Experimental signatures: Dilepton production, baryon number fluctuations, and other transport or spectroscopy observables are directly sensitive to the persistence and properties of thermoparticles and their associated spectral function features in heavy-ion collisions (Philipsen, 21 Dec 2025).

The thermoparticle concept thus unifies disparate phenomena across soft matter, mesoscopic thermodynamics, and high-temperature quantum field theory, encapsulating how thermal and non-equilibrium environments fundamentally alter the identity and collective behavior of particle-like excitations.