Spatial distributions of non-conservatively interacting particles

Non-equilibrium systems remain a frontier for physics. Indeed, even the term “non-equilibrium” applies to a large number of observed systems, and it is not immediately apparent whether there would exist a single framework which would unify all these disparate phenomena. One particular class of non-equilibrium systems, deemed “active matter”, focuses on situations where the microscopic components are being driven through continuous injection of energymarchetti_2013 . Active matter has generated special interest as continual consumption of energy microscopically is one of the characteristic properties of lifeneedleman_2017 .

Even as non-equilibrium describes a wide variety of things, active matter also applies to a very diverse variety of phenomena, existing from sub-cellular length scales to interactions among organisms. Theoretical treatments of these systems often proceed from writing down a model of the particular active system in question, usually starting from considering the forces involvedfodor_2018 ; cates_2019 . One may then either simulate the microscopic equations of motion, or coarse-grain (either with top down or bottom up approaches) the microscopic degrees of freedom into effective field treatments that capture large scale properties. These methods are usually able to be successfully compared to the real experimental realizationsbricard_2015 . Whether there exists a more generic conceptual basis for such systems is still a topic of ongoing researchtakatori_2015 ; marinibettolomarconi_2015 ; speck_2016 .

One salient difference between an active and an equilibrium system is the existence of non-vanishing microscopic currentszhang_2012 . In equilibrium all currents vanish, however for an active system this does not necessarily holdbattle_2016 . Indeed, continual consumption of energy by the microscopic components somewhat implies the existence of currents. Nevertheless, active matter can relax to a non-equilibrium steady state, where the relevant statistical properties of the system are no longer changing in time, though microscopic currents may still be present. However, while for an equilibrium system the steady state properties can be calculated from the Boltzmann distribution, which can relate the microscopic degrees of freedom to the macroscopic properties via the partition function, steady state properties of active matter cannot necessarily be calculated via this method, as they are fundamentally out of equilibrium.

We can interrogate more deeply why exactly this may be. In particular, when one writes a model for an active system it may not be immediately apparent why this microscopic description would describe a non-equilibrium system. Were one to perform a careful analysis of the equations, however, one should find that there are terms in the dynamics which would necessitate the continuous injection of energy in order to be a realistic descriptor of a physical system. The microscopic equations may, for example, break the fluctuation dissipation theorem, corresponding to the fact that there may be active noise driving the componentsdabelow_2019 . Another option is that, after one has written a microscopic description, one may see that there exists effective currents in the deterministic force part of the time evolution, a simple example could be that one microscopic degree of freedom $x_{1}$ is affecting the time evolution of another microscopic degree of freedom $x_{2}$ but that $x_{2}$ does not affect the time evolution of $x_{1}$ (or does not affect it in a commensurate way). Such a dynamical scheme would violate Newton’s third law and could only be maintained by external energy input. This simple loop would correspond to there being a non-conservative force in the system. The presence of microscopic currents at steady state also implies the existence of effective non-conservative forces in some of the microscopic degrees of freedom. A more complicated example of non-conservative forces in active matter would be the microscopic descriptions of chemically active systemsosmanovi_2019 .

The stationary state of a system where some of the forces are non-conservative can no longer be characterized as an equilibrium one, nor could the microscopic probability distribution be constructed from a Hamiltonian, as the forces in the system no longer arise as the gradient of a Hamiltonian. More broadly, the steady states of the system have to be understood from a consideration of the dynamic processes involved rather than from construction via the Boltzmann distribution.

Inspired by these ideas, in this paper we consider more generally what impacts microscopic driving, implemented as microscopic non-conservative forces, can lead to when describing collections of particles. In section II we discuss what effect a non-conservative force perturbing an equilibrium system has in terms of modifications of the ordinary Boltzmann distribution. Subsequently in section III we test these perturbation expansions for a linear system, where the microscopic probability distribution for a generic linear force field with conservative and non-conservative parts can be solved exactly using matrix methods. We then derive effective forms of the stationary distribution for arbitrary non-conservative forces. Following on from this in section IV we discuss how microscopic non-conservative forces lead to changes on macroscopic observables such as density, and demonstrate the qualitative validity of our approach against simulations of particles interacting under a non-conservative pair force. Finally, we discuss in more depth the physics that emerges from statistical ensembles of non-conservatively interacting particles in section V.

We write down the generic time evolution equations for the microscopic degrees of freedom of a system of $N$ particles evolving under a force field inside a dissipative medium:

	$\displaystyle\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}$	$\displaystyle=\frac{\mathbf{p}}{m}$		(5)
	$\displaystyle\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}$	$\displaystyle=\mathbf{F}\left(\mathbf{r}\right)-\gamma\mathbf{p}+\xi(t)$		(6)

where $\mathbf{r}$ is the vector of all the positions of particles in the system, and $\mathbf{p}$ is the vector of all the momenta in the system:

	$\displaystyle\mathbf{r}=\left(\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N}\right)$		(7)
	$\displaystyle\mathbf{p}=\left(\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{N}\right)$		(8)

We have employed the assumption for this model that the force field $\mathbf{F}$ depends on the position degrees of freedom of the system only $\mathbf{F}$. This assumption could in principle be relaxed in order to describe different classes of active systems where the forces depend on the momenta. We also assume that these particles exist in a media with a dissipative term $\gamma$ and a white noise term $\xi$ which has the property that $\langle\epsilon(t)\epsilon(t^{\prime})\rangle=2k_{b}T\gamma\delta(t-t^{\prime})$.

We now restrict ourselves to analysis of the stationary distribution of systems with this time evolution. We further focus our attention on the case where the system is overdamped. We wish to obtain knowledge of the microscopic probability of as a function of the positions of all the particles $P(\mathbf{r})$. The corresponding Smoluchowski equation for the evolution of the probability density under these forces is given byrisken_1991 :

$$\frac{\partial P(\mathbf{r},t)}{\partial t}=\nabla.\left(-\mathbf{F}\left(\mathbf{r}\right)P(\mathbf{r},t)+k_{b}T\nabla P(\mathbf{r},t)\right)$$

(9)

Where we have introduced the $\nabla$ differential operator, which is given by $\nabla=\left(\frac{\partial}{\partial\mathbf{r}_{1}},\frac{\partial}{\partial\mathbf{r}_{2}},\ldots,\frac{\partial}{\partial\mathbf{r}_{N}}\right)$. As the probability must be greater than or equal to zero, we represent the probability density as $P(\mathbf{r})=\exp(-\phi(\mathbf{r}))$. The stationary state distribution is the one in which the time derivative of the probability is zero, in other words, the left hand side of equation 9 is equal to zero. From equation 9, the stationary probability density in this system must obey the following relationship:

$$\nabla.\left(\exp\left(-\phi(\mathbf{r})\right)\left(-\mathbf{F}(\mathbf{r})-k_{b}T\nabla\phi(\mathbf{r})\right)\right)=0$$

(10)

One can see trivially if the forces in the system arise from the gradient of a function (such as the Hamiltonian) $F(\mathbf{r})=-\nabla H(\mathbf{r})$ that the stationary probability distribution over microstates will be given by the Boltzmann distribution $P(\mathbf{r})=\exp(-H(\mathbf{r})/k_{b}T)$. For systems in which the forces do not arise as the gradient of a function, the solutions are not so trivial. We remind readers that we use the terms conservative to describe forces which are expressible as the gradient of a scalar function, and non-conservative to describe forces which cannot be expressed in this way. Following Risken and othersrisken_1991 ; wedemann_2016 , we imagine that the force can be split into two components:

$$F(\mathbf{r})=\mathbf{f}^{(c)}(\mathbf{r})+\mathbf{f}^{(a)}(\mathbf{r})$$

(11)

Where the new conditions that have to be specified for the probability distribution to be stationary are given by:

	$\displaystyle\mathbf{f}^{(c)}(\mathbf{r})$	$\displaystyle=-k_{b}T\nabla\phi(\mathbf{r})$		(12)
	$\displaystyle\nabla.\left(e^{-\phi(\mathbf{r})}\mathbf{f}^{(a)}(\mathbf{r})\right)$	$\displaystyle=0$		(13)

However, this is only a redefinition of the problem equation 10. Were we to be able to find the proper splitting of the force fields we could in principle solve for the stationary distribution. We will now focus our attention on an example system given by forces:

$$\mathbf{F}(\mathbf{r})=-\nabla H_{0}(\mathbf{r})+\mathbf{m}(\mathbf{r})$$

(14)

Where $H_{0}$ defines some equilibrium Hamiltonian, but we keep $\mathbf{m}(\mathbf{r})$ to be as general as possible, such that it may include conservative and non-conservative elements. We here note that the splitting operation defined in equations 12 and 13 is not the same as merely separating the conservative and non-conservative components of the force, as can be seen from equation 13, which will not generally be equal to zero even if $\nabla.\mathbf{m}(\mathbf{r})=0$. The challenge is therefore to find the splitting of the force field which satisfies these conditions. This is a difficult problem, but can be recast into the form of a differential equation by making the following addition, introducing the new scalar field $\chi$:

	$\displaystyle f^{(c)}(\mathbf{r})$	$\displaystyle=-\nabla H_{0}(\mathbf{r})-\nabla\chi(\mathbf{r})$		(15)
	$\displaystyle f^{(a)}(\mathbf{r})$	$\displaystyle=m(\mathbf{r})+\nabla\chi(\mathbf{r})$		(16)

this trivially satisfies the conservative condition of the log probability equation 12. In order to find the probability distribution we are then left with satisfying the following differential equation for $\chi$ from equation 13:

$$\nabla^{2}\chi(\mathbf{r})-\beta(\nabla H_{0}(\mathbf{r})+\mathbf{m}(\mathbf{r})).\nabla\chi(\mathbf{r})-\beta(\nabla\chi(\mathbf{r}))^{2}=\beta\nabla H_{0}(\mathbf{r}).\mathbf{m}(\mathbf{r})-\nabla.\mathbf{m}(\mathbf{r})$$

(17)

where we have introduced the notation that $k_{b}T=\beta^{-1}$. This is a $N$ dimensional non-linear partial differential equation in the unknown $\chi(\mathbf{r})$, which is by itself still a challenging proposition. However if we define the following substitution $\chi(\mathbf{r})=-\beta^{-1}\log\left(\mu(\mathbf{r})\right)$ we are left with the following linear equation in $\mu(\mathbf{r})$:

	$\displaystyle\left(\nabla^{2}-\mathbf{q_{1}}(\mathbf{r}).\nabla+q_{2}(\mathbf{r})\right)\mu(\mathbf{r})=0$		(18)
	$\displaystyle\mathbf{q_{1}}(\mathbf{r})=\beta(\nabla H_{0}(\mathbf{r})+\mathbf{m}(\mathbf{r}))$		(19)
	$\displaystyle q_{2}(\mathbf{r})=\beta^{2}\nabla H_{0}(\mathbf{r}).\mathbf{m}(\mathbf{r})-\beta\nabla.\mathbf{m}(\mathbf{r})$		(20)

Where only solutions with $\mu(\mathbf{r})>0$ are physical ($\chi(\mathbf{r})$ is real). We assume a perturbative expansion exists in the magnitude of the additional forces $\mathbf{m}$. Parameterizing the additional forces with a parameter $\epsilon$:

$$\mathbf{m}\to\epsilon\mathbf{m}$$

(21)

and seeking solutions of $\mu(\mathbf{r})$ of the form $\mu(\mathbf{r})\approx\sum_{n=0}\epsilon^{n}\mu_{n}(\mathbf{r})$.

Substitution of this expansion leads to the following differential equation for the $nth$ order perturbation:

$$\left(\nabla^{2}-\beta\nabla H_{0}(\mathbf{r}).\nabla\right)\mu_{n}(\mathbf{r})=(\beta\mathbf{m(r)}.\nabla-q_{2}(\mathbf{r}))\mu_{n-1}(\mathbf{r})$$

(22)

Where the expression on the left is a linear operator acting on $\mu_{n}(\mathbf{r})$ and the term on the right is some forcing term. This series would be supplemented by the conditon that $\mu_{0}=\text{const}$. The equation to each individual $\mu_{n}$ is acted upon by the same linear operator for every $n$, the only differences are the source terms on the right hand side of the equations. Therefore, were one to be able to find a Green’s function of the linear operator acting on each individual $\mu_{n}$, given by the following equation:

$$\left(\nabla^{2}-\beta\mathbf{\nabla}H_{0}.\nabla\right)G(\mathbf{r},\mathbf{r^{\prime}})=\delta(\mathbf{r-r^{\prime}})$$

(23)

with the appropriate boundary conditions that $G$ vanishes at infinity. Then one may express the entire series solution in terms of this function as:

	$\displaystyle\mu_{0}(\mathbf{r})$	$\displaystyle=\text{const}$		(24)
	$\displaystyle\mu_{1}(\mathbf{r})$	$\displaystyle=-\mu_{0}\int G(\mathbf{r},\mathbf{r^{\prime}})q_{2}(\mathbf{r^{\prime}})d\mathbf{r^{\prime}}$		(25)
	$\displaystyle\vdots$			(26)
	$\displaystyle\mu_{n}(\mathbf{r})$	$\displaystyle=\int G(\mathbf{r},\mathbf{r^{\prime}})(\beta\mathbf{m(r^{\prime})}.\nabla^{\prime}-q_{2}(\mathbf{r^{\prime}}))\mu_{n-1}(\mathbf{r^{\prime}})\mathrm{d}\mathbf{r^{\prime}}$		(27)

As the Green’s function only depends on the equilibrium properties of the system this representation shows the effect of non-conservative force fields in terms of integrals over effective source terms arising from the imposition of the non-conservative force.

This series would be the steady state solution of the microscopic probability for a system evolving under a generic force field. However, the complexity of this representation is hardly less than the original equation, given the difficulty in evaluating the coordinate space Green’s functions. We will revisit this question in a later section and for now also discuss the physical significance of the perturbations. For completeness, we note that the first order perturbation is given by:

$$P(\mathbf{r})\approx\exp\left(-\beta H_{0}(\mathbf{r})+\epsilon\beta\int\mathrm{d}\mathbf{r^{\prime}}G(\mathbf{r},\mathbf{r^{\prime}})\left(\beta\nabla^{\prime}H_{0}(\mathbf{r^{\prime}}).\mathbf{m(\mathbf{r^{\prime}})}-\nabla.\mathbf{m}(\mathbf{r^{\prime}})\right)\right)$$

(28)

Which is already a familiar result in the recent literaturemoyses_2015 ; noh_2015 , though we here display it in terms of the Green’s function.

We have in the previous section displayed the coordinate space probability of a non-conservative system in terms of a perturbative expansion in the non-conservative forces. However, the form of the distribution is very complicated for the purposes of calculation. As we are not so much interested in exact mathematical results as opposed to leading order physical effects, in this section we show the validity of the perturbative approach for a linear system (where exact results can be obtained) by comparing how the first order theory compares to the exact solution. We then discuss in more depth the coordinate space Green’s functions which we introduced in the previous section, and approximate forms of the microscopic probability distribution.

The preceding sections were rather abstract in their application, of more immediate importance in experimental realizations would be the larger scale properties of such systems. In this section we will try to more phenomenologically understand how microscopic driving will lead to macroscopic effects. The correct macroscopic degrees of freedom for non-equilibrium system are not obvioussolon_2015 , so we will choose to focus on the density of the particles, which always exists and is well defined in and out of equilibrium. Rather than treating the density as a conserved field and reasoning over the form of the density currents from physics in a continuity equation like scheme, we instead wish to focus on how our microstate probability 45 would modify the macroscopic density that one would observe for any non-conservative system.

We introduce by hand such systems interacting with both conservative and non-conservative parts by introducing an extra pair force between particles which is given by a curl of some vector $\mathbf{m_{ij}}(\mathbf{r}_{ij})=\nabla_{i}\times\mathbf{A}(\mathbf{r}_{ij})$, but stress these are not meant to represent real physical systems. In particular by choosing a pair non-conservative force arising in this way we have chosen axes of the system. The closest physical picture that would correspond to this kind of system would be one in which a global field (such as a magnetic field) induces a constant angular momentum in the particles relative to the imposed field, and then these angular momenta interact with each other according to some effective curl, reminiscint of some experimentally realized systemssoni_2019 . In order to model real active matter systems, we would have to introduce particles with their own orientational degrees of freedom, Alternatively, non-conservative three body forces between particles could be introduced. This notwithstanding, these models can show us qualitatively the effect that non-conservative microscopic forces have on spatial distributions of the system. We are particularly interested in whether any generalities arise from the consideration of the modified Boltzmann distribution in equation 45.

As we mentioned, the observable quantity we are most interested in is the density of a fluid of particles interacting non-conservatively, which corresponds to the statistical average of the density operator under the stationary distribution:

	$\displaystyle\rho(\mathbf{x},\mathbf{r})$	$\displaystyle=\sum_{i=1}^{N}\delta(\mathbf{x}-\mathbf{r_{i}})$		(46)
	$\displaystyle\langle\rho(\mathbf{x})\rangle$	$\displaystyle=\int\mathrm{d}\mathbf{r}\rho(\mathbf{x},\mathbf{r})P(\mathbf{r});$		(47)

where $\mathbf{x}$ now refers to ordinary three dimensional space. In particular we are interested in how densities differ for systems with non-conservative microscopic forces. In equilibrium there is a well defined connection between the free energy of the system and this density, which is captured by a free energy functionalevans_1979 :

$$F\left[\rho(\mathbf{x})\right]=\beta^{-1}\int\mathrm{d}\mathbf{x}\rho(\mathbf{x})\left(\log(\lambda_{TH}^{3}\rho(\mathbf{x}))-1\right))+\frac{1}{2}\iint\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{x^{\prime}}\rho(\mathbf{x})\rho(\mathbf{x^{\prime}})U(\mathbf{x}-\mathbf{x^{\prime}})$$

(48)

Where $U$ is some pair potential acting between the particles. The second term of this equation, arising from interparticle potentials, can be derived from reformulating the Hamiltonian $H=\sum_{i\neq j}^{N}U(\mathbf{r_{i}}-\mathbf{r_{j}})$ using the density operator. The density field which minimizes this functional is the best approximation to the true density of a system of particles interacting with the potential $U$.

We can reason about what the effect of pairwise non-conservative driving is by considering the effective source term that appears in the modified microscopic probability $\nabla H_{0}.\mathbf{m}$. There are several salient facts about this source term, the most important of which is that if it is equal to zero, either from the non-conservative force being zero, or the angle between the non-conservative force and the conservative force is always equal to zero, then the stationary properties of the system will be given by a Boltzmann distribution arising from the conservative part. This source can also be described using the density operator in the same way as the equilibrium Hamiltonian, using pairwise addititive curl force between particles (ignoring density fluctuations):

$$\nabla H_{0}.\mathbf{m}\approx\iiint\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{x^{\prime}}\mathrm{d}\mathbf{x^{\prime\prime}}\rho(\mathbf{x})\rho(\mathbf{x^{\prime}})\rho(\mathbf{x^{\prime\prime}})\left(\nabla U(\mathbf{x}-\mathbf{x^{\prime}}).\nabla\times A(\mathbf{x}-\mathbf{x^{\prime\prime}})\right)$$

(49)

Where now the differential operators $\nabla$ are with respect to ordinary 3 dimensional space.

In the previous section, we saw that the first order approximation for the microscopic probability included an additional term in the exponent of the Boltzmann distribution of the form $\nabla H_{0}.\mathbf{m}$. We introduce a qualitatively the effect of this driving by introducing the following functional:

	$\displaystyle F^{\text{(nc)}}=$	$\displaystyle\beta^{-1}\int\mathrm{d}\mathbf{x}\rho(x)\left(\log(\lambda_{TH}^{3}\rho(\mathbf{x}))-1\right))+\frac{1}{2}\iint\mathrm{d}\mathbf{x^{\prime}}\mathrm{d}\mathbf{x}\rho(\mathbf{x})\rho(\mathbf{x^{\prime}})U(\mathbf{x}-\mathbf{x^{\prime}})$		(50)
		$\displaystyle+\lambda_{1}^{2}\beta\iiint\mathrm{d}\mathbf{x^{\prime\prime}}\mathrm{d}\mathbf{x^{\prime}}\mathrm{d}\mathbf{x}\rho(\mathbf{x})\rho(\mathbf{x^{\prime}})\rho(\mathbf{x^{\prime\prime}})\nabla U(\mathbf{x}-\mathbf{x^{\prime}}).\nabla\times\mathbf{A}\left(\mathbf{x}-\mathbf{x^{\prime\prime}}\right)$		(51)

While we label this $F$, we wish to be careful in calling this a free energy, as this system is inherently not in equilibrium. However we assume a similar mapping exists from a qualitative description of the microscopic probability in our non-conservative case to an effective functional of the particle density (with all the subtleties this entails). The first term of this functional is the familiar ideal gas contribution. The next encompasses the (conservative) pairwise energy between particles in our system. The next corresponds to modifications due to the non-conservative forces, which exist as an effective three body potentialiyetomi_1989 of special form.

While the contribution to the free energy due to interparticle potentials is fairly unconstrained, the term corresponding to non-conservative pair forces is more interesting. A salient mathematical fact of integrals of this type is that:

$$\int_{\text{all space}}\mathrm{d}\mathbf{x}\nabla U(\mathbf{x}).\nabla\times\mathbf{A}(\mathbf{x})=0$$

(52)

Therefore the contribution to this functional arising from any non-conservative forces will always be equal to 0 when the density is constant. Taking this logic further, pairwise non-conservative interactions, if they do anything to the density, would be to drive the density profiles of the system from the homogeneuous state. For real physical systems being internally driven, we would have to consider higher order terms in order to capture the relevant physics, however a similar picture would emerge for terms which depend on this source squared and so on. This interesting result arises solely from the fact that the forces are non-conservative, and decomposable in terms of curls.

We can see how this effective functional changes for particular kinds of density field $\rho(\mathbf{x})$. If the source has a finite length scale and the density is approximately constant over this length scale, then the contributions to $F^{(}{nc})$ from these terms are effectively zero (ignoring fluctuations). Therefore, the main contribution to $F^{(nc)}$ from non-conservative forces happens due to surfaces. This can be proven using the divergence theorem for a constant but finite density field, however we can demonstrate this using simulations with pairwise non-conservative forces. In order to test this we simulate 10000 particles in a box with periodic boundary conditions interacting with both an ordinary Lennard-Jones potential:

$$U(r)=4\epsilon_{LJ}\left(\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right)$$

(53)

and where we now introduce an extra divergence-free force between the particles, again not meant to represent a real physical system, but rather as a qualitative representation for how non-conservative forces affect spatial density:

$$\mathbf{F}_{ij}(x_{ij},y_{ij},z_{ij})=\frac{\epsilon}{\lambda^{2}}\nabla_{i}\times\frac{1}{\lambda^{2}}\left(\left(x_{ij},y_{ij},z_{ij}\right).f(s_{i},s_{j}).\left(x_{ij},y_{ij},z_{ij}\right)\exp(-(x_{ij}^{2}+y_{ij}^{2}+z_{ij}^{2})/\lambda^{2})\right)$$

(54)

Where $f(s_{i},s_{j})$ would correspond to a matrix that depends on some microscopic properties of the particles (such as orientation $s$). Instead of treating these internal degrees of freedom fully in a simulation, we fix the form of the matrix with respect to the system coordinates, somewhat like imposing a strong field aligning the particles. In order to show qualitative variation from expected equilibrium density profiles, we first simulate the relaxation of the system to an equilibrium state, and then switch on the non-conservative forces. In addition, as we expect non-conservative forces to mainly affect surfaces, we keep the total particle volume fraction of the system small $\frac{N\pi\sigma^{3}}{6V}=0.01$. The other parameters we choose are that $k_{b}T=1$, $\lambda=1.5$, $\sigma=1$,$\epsilon_{LJ}=2.0,\epsilon=1$

We show a particular example of the shape adopted by this system in figure 3. This simple simulation already shows qualitatively the effects that we were able to reason about through a consideration of the source term. The extra term in the effective “free energy” drives the system away from its equilibrium shape (which would be a spherical droplet), leading to the formation of a pancake like shape. It should be noted that while this appears to be similar to elastic deformation of a sphere, the system is heavily overdamped and the states arise from microscopic pair forces. From the preceding discussion we stated that the source term $\nabla H_{0}.m$ would act primarily through surfaces. We can now quantify this effect by looking at the same snapshots of system evolution with the color scale of the particles equal to their total value of this source and also see how the average value of this term changes during our simulation run.

These results are presented in figure 4. Several trends are visible from this figure. Firstly, that the mean value of $\nabla H_{0}.\mathbf{m}$ per particle changes during the simulation as it moves from the spherical shape to the pancake shape. However, the distribution of $\nabla H_{0}.\mathbf{m}$ over all the particles is very broad compared to the mean. Moreover, these values are not distributed equally across the final state, but are instead localized in different parts of the pancake. As can be seen, there exists a strong surface layer across one axes of the pancake, and a differently valued one across another surface, according well with the mathematical arguments which we presented earlier.

This simple phenomenology and simulations show us qualitatively the effect non-conservative microscopic interactions can have on larger scale observations. We kept our discussion general up until the point of simulating the particles. Interestingly, a natural result of a framework considering the interplay of microscopic non-conservative interactions was that it will subsequently manifest itself through density inhomogeneities, and most strongly through surfaces. In particular, the imposition of given forms of microscopic driving naturally lead to particular shapes when considering aggregates of such particles, given by the interplay of the conservative and non-conservative forces. In this system the non-conservative and conservative forces compete against each other over which shape they find preferable, settling on the pancake shown.

We have in the preceding displayed results relating to both the microscopic and macroscopic treatments of non-conservative forces in a system of particles. We will now proceed to discuss some broader aspects of the physical relevance of these results, in both its microscopic and macroscopic manifestations.

In this paper, we have analyzed the steady state distribution that arise from particles interacting with both conservative and non-conservative forces. We have shown that the Boltzmann distribution is modified by the addition of excess convolution integrals over effective source terms that arise when treating the system pertubatively in the non-conservative elements. Finally, we give some indication for how non-conservative forces affect more macroscopic observables, and find that generic non-conservative forces couple to density surfaces rather than volume terms. The consequences of this are that microscopic driving implemented between pairs of particles leads to the adoption of different density profiles and shapes than would be expected from the same system with only the conservative interactions. However, the fact that such deformations arise can be understood from the source terms we derived in the microscopic theory, where differences from Boltzmann distributions arise due to operators acting on $\nabla H_{0}\left(\mathbf{r}\right).m\left(\mathbf{r}\right)$, and this perturbation, when course grained, naturally couples to surfaces. We believe the framework we have presented so far is suggestive of future directions.

On the mathematical level, the full microscopic probability distribution is very complicated. The probability distribution due to non-conservative forces can be written as a series of integrals over source terms with Green’s functions, where the Green’s function is some equilibrium path integral. Ignoring this path integral and treating the effects of non-conservative forces as a local modification of probability was seen to approximately reproduce real probabilities, and was also qualitatively capture features on the larger scale.

While in this manuscript we have mainly focused on the static distributions of matter being driven by non-conservative forces, a whole other aspect that was ignored in this work was the dynamics. The steady states arising from these forms of non-equilibrium systems may still have net currents acting. How these steady states then behave when perturbed by an external field, their response properties and dynamic relaxation are all ongoing topics of researchmaes_2008 . An additional assumption we employed was that the system was overdamped. Upon relaxation of this assumption many other interesting phenomena may result.

In the latter sections, we noted that non-conservative systems adopt particular shapes compared to conservative ones. We believe this is an interesting avenue of potential exploration. In particular, one may be able to relate particular mesoscopic patterns or shapes to the interplay of microscopic conservative and non-conservative forces. Additionally, one may be able to reformulate this theory in terms of conservative and non-conservative stress tensors krger_2018 .

The author would like to the thank Hridesh Kedia, Peter Foster and Yitzhak Rabin for helpful discussions. This work was funded through the Gordon and Betty Moore foundation.