# Slow and fast particles in shear-driven jamming: critical behavior and
finite size scaling

Peter Olsson  Department of Physics, Umeå University, 901 87 Umeå, Sweden

###### Abstract

We do shear-driven simulations of a simple model of non-Brownian particles in
two dimensions. By examining the velocity distribution at different densities
and shear rates we find strong evidence for the existence of two different
processes, respectively dominated by the slower and the faster particles—the
slow process and the fast process. The leading divergence in the shear
viscosity is governed by the fast process. An examination of height and
position of the low-velocity peak in the distribution demonstrates that it is
the slow process that is responsible for the correction-to-scaling term in the
critical scaling analysis. We further find that the presence of velocity
correlations across large distances is primarily due to the slow process which
implies that the diverging viscosity and the diverging correlation length are
only indirectly related.

###### pacs:

63.50.Lm, 45.70.-n 83.10.Rs

Particle transport is ubiquitus in both industry and every-day life with
varying behaviors due e.g. to differing particle properties and geometries. A
reasonable approach in the quest for a better understanding of the slowing
down of the dynamics, e.g. because of an increase in density, is to first
examine simplified models. One of the simplest possible consists of a
collection of circular disks in two dimensions with contact-only interactions
O’Hern  _et al._ (2003). When such a collection of particles is driven at a
constant shear strain rate $\dot{\gamma}$ Durian (1995), the system develops a
shear stress $\sigma$, and the shear viscosity,
$\eta\equiv\sigma/\dot{\gamma}$, diverges as the jamming density, $\phi_{J}$,
is approached from below. For the conceptually simple case of hard disks and
overdamped dynamics this is seen in an algebraic divergence,
$\eta\sim(\phi_{J}-\phi)^{-\beta}$. This divergence is related to the increase
in contact number $z$ towards the isostatic value,
$z_{c}-z\sim(\phi_{J}-\phi)^{u_{z}}$. In the thermodynamic limit the isostatic
contact number is $z_{c}=2d$ Alexander (1998); see Ref. Goodrich  _et al._
(2012) for the generalization to finite $N$. In terms of the contact number
deficit the divergence becomes $\eta\sim(z_{c}-z)^{-\beta/u_{z}}$.

A hallmark of critical phenomena is a diverging spatial correlation length. It
has also long been realized that particle motion in sheared systems becomes
increasingly collective as the jamming density is approached Pouliquen (2004);
Lechenault  _et al._ (2008); Heussinger  _et al._ (2010); Hexner  _et al._
(2018). In a recent analysis of the velocity field Olsson and Teitel (2020) it
was further found that fluctuations in the rotation and the divergence of the
velocity field behave differently, and that the length related to the
rotations appears to be the more important one, diverging as
$\xi\sim(\phi_{J}-\phi)^{-\nu}$, with $\nu\approx 1$ Olsson and Teitel (2020).

Because of difficulties with numerically simulating the dynamics of hard
particles, simulations are commonly performed with elastic particles with
forces related to the particle overlaps. The shear viscosity has then a strong
shear strain rate dependence—see Fig. 1(a)—and in attempted critical scaling
analyses Olsson and Teitel (2007, 2011) it has furthermore become clear that
one also needs to include a correction-to-scaling term Olsson and Teitel
(2011); Kawasaki  _et al._ (2015). In two dimensions (2D) such determinations
of $\beta$ have typically given values in the range $\beta=2.2$ through 2.83
Andreotti  _et al._ (2012); Olsson and Teitel (2011); Kawasaki  _et al._
(2015). For the combination $\beta/u_{z}$ different methods that directly
probe the hard disk limit give $\beta/u_{z}=1/0.38=2.63$ Lerner  _et al._
(2012) and $\beta/u_{z}=2.69$ Olsson (2015). (Determinations in three
dimensions tend to give higher values, $\beta/u_{z}\approx 3.3$ DeGiuli  _et
al._ (2015) or $\beta/u_{z}=3.7\pm 0.7$ Olsson (2019). An attempt to explain
this dependence on dimensionality in terms of a finite size effect is
discussed in Nishikawa  _et al._ (2021); Olsson (2022a).)

From analytical considerations the exponent has however been argued to be
$\beta/u_{z}\approx 3.41$ DeGiuli  _et al._ (2015); Ikeda (2020) and it has
then been claimed that the determinations quoted above for 2D are incorrect
due to the neglect of logarithmic corrections to scaling DeGiuli  _et al._
(2015); Ikeda (2020); Nishikawa  _et al._ (2021). Though this explanation is a
possibility, it could also be that the discrepancy only points to a lack of
understanding of the phenomenon of shear-driven jamming.

In the present Letter we present evidence for a novel picture that describes
shear-driven jamming as being controlled by two different processes dominated
by the slow and the fast particles, respectively, and accordingly coined the
“slow process” and the “fast process”. The fast process is responsible for the
leading term in the divergence of the shear viscosity and is dominated by
particles in the tail of the velocity distribution Olsson (2016). The slow
process leads to the correction-to-scaling term Olsson and Teitel (2011);
Kawasaki  _et al._ (2015), and arises from particles at and below the low-
velocity peak in the distribution. It is further found that the presence of
velocity correlations across large distances is related to the slow process
whereas the fast process appears to be short range correlated, only. The
present Letter gives a short description of a comprehensive examination of
shear-driven jamming; a more detailed discussion is given in Ref. joi (2022),
except for the finite size dependence, which will be discussed elsewhere
Olsson (2022b). The analyses presented here work the same also in three and
four dimensions, presentations of these results will however also be deferred
to a later publication.

We simulate a bidisperse collection of particles in 2D with equal number of
particles of two different sizes. The total number of particles is $N=65536$
particles, if not otherwise noted. The small particles have diameter $d_{s}=1$
and the size ratio is 1.4 O’Hern  _et al._ (2003). For particles in contact we
define the relative overlap $\delta_{ij}=1-r_{ij}/d_{ij}$ where $r_{ij}$ is
the distance between particles $i$ and $j$ and $d_{ij}$ is the sum of their
radii. The contact interaction is from the potential energy
$V_{p}(r_{ij})=\epsilon\delta^{2}_{ij}/2$; we take $\epsilon=1$. The force on
particle $i$ from particle $j$ is
$\mathbf{f}^{\mathrm{el}}_{ij}=-\nabla_{i}V_{p}(r_{ij})$, which gives
$f^{\mathrm{el}}_{ij}=\epsilon\delta_{ij}/d_{ij}$. We do shearing simulations
with a time-dependent shear strain $\gamma=\dot{\gamma}t$ and Lees-Edwards
boundary conditions Evans and Morriss (1990) on a system with volume
$V=L\times L$. The shearing gives an average homogenous velocity profile
$=\dot{\gamma}y\hat{x}$ but our focus will be on the non-affine velocity,
which is the particle velocity relative to this velocity profile,
$\mathbf{v}_{i}=\mathbf{v}^{\mathrm{tot}}_{i}-\dot{\gamma}y_{i}\hat{x}$.
Related to the non-affine velocity is the dissipative force
$\mathbf{f}^{\mathrm{dis}}_{i}=-k_{d}\mathbf{v}_{i}$. We simulate with
overdamped dynamics such that
$\mathbf{f}^{\mathrm{el}}_{i}+\mathbf{f}^{\mathrm{dis}}_{i}=0$ which becomes
$\mathbf{v}_{i}=\mathbf{f}^{\mathrm{el}}_{i}/k_{d}$. We take $k_{d}=1$ and the
unit of time $\tau_{0}=d_{s}^{2}k_{d}/\epsilon=1$. The shear stress is
$\sigma=-\left<\mathbf{p}^{\mathrm{el}}_{xy}\right>$ from the pressure tensor
which is obtained from the forces between the contacting particles,
$\mathbf{p}^{\mathrm{el}}=V^{-1}\sum_{i<j}\mathbf{f}^{\mathrm{el}}_{ij}\otimes\mathbf{r}_{ij}$.
Because of the large $N$ in our simulations the fluctuations in pressure
$\equiv\frac{1}{2}(\mathbf{p}^{\mathrm{el}}_{xx}+\mathbf{p}^{\mathrm{el}}_{yy})$
during the run are very small.

Figure 1: The divergence of the shear viscosity. Panel (a) is the shear
viscosity $\eta\equiv\sigma/\dot{\gamma}$ for different shear strain rates.
The vertical dashed line is $\phi_{J}$ and the solid line is the approximate
critical divergence $\eta(\phi,\dot{\gamma}\to
0)\sim(\phi_{J}-\phi)^{-\beta}$, $\beta=2.7$. Panel (b) which is $\sigma$ vs
$\dot{\gamma}$ at $\phi=0.8434\approx\phi_{J}$ illustrates the determination
of the exponents $q$ and $q_{2}$ defined in Eq. (3).

The analyses below will be done in terms of the shear stress $\sigma$ but to
illustrate the jamming transition Fig. 1(a) shows
$\eta(\phi,\dot{\gamma})\equiv\sigma(\phi,\dot{\gamma})/\dot{\gamma}$ for
shear strain rates $\dot{\gamma}=10^{-8}$ through $2\times 10^{-5}$. The
transition is shown by the rapid increase of $\eta$ with $\phi$, which in the
$\dot{\gamma}\to 0$ limit approaches $\eta\sim(\phi_{J}-\phi)^{-\beta}$, with
$\beta\approx 2.7$, illustrated by the solid line. The analyses of this kind
of data (and the similar $\eta_{p}=p/\dot{\gamma}$) in the literature Olsson
and Teitel (2007, 2011) rely on the standard scaling assumption Vågberg  _et
al._ (2016),

$\sigma(\phi,\dot{\gamma})b^{y/\nu}=\bar{g}_{\sigma}(\delta\phi\,b^{1/\nu},\dot{\gamma}b^{z})+b^{-\omega}\bar{h}_{\sigma}(\delta\phi\,b^{1/\nu},\dot{\gamma}b^{z}).$
(1)

Here $b$ is a length rescaling factor, $y$ is the scaling dimension of
$\sigma$, $\nu$ is the correlation length exponent,
$\delta\phi=\phi-\phi_{J}$, $z$ is the dynamical exponent, $\omega$ is the
correction-to-scaling exponent and $\bar{g}_{\sigma}$ and $\bar{h}_{\sigma}$
are unknown scaling functions. With $b=\dot{\gamma}^{-1/z}$ in Eq. (1)
together with $q=y/z\nu$ and $q_{2}=q+\omega/z$ one finds

$\sigma(\phi,\dot{\gamma})=\dot{\gamma}^{q}g_{\sigma}\left(\frac{\phi-\phi_{J}}{\dot{\gamma}^{1/z\nu}}\right)+\dot{\gamma}^{q_{2}}h_{\sigma}\left(\frac{\phi-\phi_{J}}{\dot{\gamma}^{1/z\nu}}\right).$
(2)

In the fitting of Ref. Olsson and Teitel (2011) the scaling functions were
taken to be exponentials of polynomials in
$(\phi-\phi_{J})/\dot{\gamma}^{1/z\nu}$, and the parameters of these
polynomials together with $\phi_{J}$ and the critical exponents were adjusted
to get the best possible fit. [Since $\omega>0$, as it is an irrelevant
scaling variable, it follows that $q_{2}>q$, and that for $\eta$ the two terms
of Eq. (2) scale as $\dot{\gamma}^{q-1}$ and $\dot{\gamma}^{q_{2}-1}$, which
implies that the first term is the more divergent one as $\dot{\gamma}\to 0$.]

As shown in Eq. (2) the scaling expression for $\sigma(\phi,\dot{\gamma})$
consists of two different terms, as first reported in Ref. Olsson and Teitel
(2011), and the question that we set out to answer in the present Letter is
the physical mechanisms behind these two terms. To simplify the analyses we
here focus on the behavior at the jamming density $\phi=\phi_{J}$. An
alternative, which is to examine the simulations in the hard disk limit, is
shown in Sec. III E of Ref. joi (2022). At $\phi=\phi_{J}$ Eq. (2) simplifies
to,

$\sigma(\phi_{J},\dot{\gamma})=a_{1}\dot{\gamma}^{q}+a_{2}\dot{\gamma}^{q_{2}}.$
(3)

With $\phi_{J}\approx 0.8434$ Heussinger and Barrat (2009); Olsson and Teitel
(2011) and with the methods described in Ref. joi (2022) we obtain the
exponents $q=0.284(2)$ and $q_{2}=0.567(7)$, in good agreement with previous
works Olsson and Teitel (2011). The prefactors are $a_{1}=0.00437$ and
$a_{2}=0.067$. The fit is shown in Fig. 1(b).

Figure 2: Velocity distributions at $\phi=0.8434\approx\phi_{J}$. Panel (a)
gives ${\cal P}(v)$ for several different shear strain rates with symbols as
in Fig. 1(a). Each data set has a clear peak and panel (b) shows the same data
rescaled to make the peaks coincide. It is then found that the rescaled ${\cal
P}(v)$ collapse below and up to the peak whereas the data above the peak
depend strongly on $\dot{\gamma}$. At high velocities the distributions decay
algebraically with a $\dot{\gamma}$-dependent exponent. Panels (c) and (d)
show the algebraic dependences of peak height and position on $\dot{\gamma}$:
${\cal P}_{p}\sim\dot{\gamma}^{u_{\cal P}}$ and
$v_{p}\sim\dot{\gamma}^{u_{v}}$.

To examine the shear stress in a novel way we turn to the properties of the
velocity distribution ${\cal P}(v)$ calculated from the non-affine velocities
$v_{i}\equiv|\mathbf{v}_{i}|$, normalized such that $\int{\cal P}(v)dv=1$.
Fig. 2(a) shows ${\cal P}(v)$ vs $v$ at $\phi=0.8434\approx\phi_{J}$ for
$\dot{\gamma}=10^{-8}$ through $2\times 10^{-5}$. For each shear strain rate
there is a peak at $v=v_{p}$ with peak height ${\cal P}_{p}={\cal P}(v_{p})$.
A key observation is now that the data up to and slightly above the peak
collapse to a common function, as shown in Fig. 2(b), when ${\cal P}(v)$ and
$v$ are rescaled to make the peaks fall on top of each other. The same is true
also for ${\cal P}(v)$ from a range of densities both below and above
$\phi_{J}$ (see joi (2022)). At higher velocities ${\cal P}(v)$ is algebraic,
${\cal P}(v)\sim v^{-r}$, with an exponent $r$ that varies with $\phi$ and
$\dot{\gamma}$. [Earlier analyses suggest that $r\to 3$ as jamming is
approached Olsson (2016). The distributions are eventually cut off
exponentially at large $v$.]

We now set out to show that the second term in Eq. (3) is related to the peak
in ${\cal P}(v)$. With the well known power balance
$\sigma\dot{\gamma}=(N/V)k_{d}\left<v^{2}\right>$, which is a relation between
input power $V\sigma\dot{\gamma}$ and dissipated power
$Nk_{d}\left<v^{2}\right>$ Ono  _et al._ (2003), the shear stress may be
written in terms of ${\cal P}(v)$ as

$\sigma=\frac{N}{V}\frac{k_{d}}{\dot{\gamma}}\int{\cal P}(v)v^{2}dv.$ (4)

Introducing $x=v/v_{p}$ and $f(x)={\cal P}(v)/{\cal P}_{p}$, the contribution
to $\sigma$ from velocites up to the peak becomes

$S(v_{p})=\frac{N}{V}k_{d}W_{p}\int_{0}^{1}f(x)x^{2}dx,$ (5)

where $W_{p}={\cal P}_{p}v_{p}^{3}/\dot{\gamma}$. Fig. 2(c) and (d) show that
peak height and position depend algebraically on $\dot{\gamma}$: ${\cal
P}_{p}\sim\dot{\gamma}^{u_{{\cal P}}}$, with $u_{{\cal P}}=-0.733$, and
$v_{p}\sim\dot{\gamma}^{u_{v}}$, with $u_{v}=0.766$. For the
$\dot{\gamma}$-dependence of $S(v_{p})\sim W_{p}$ we then find

$W_{p}(\phi_{J},\dot{\gamma})\sim\dot{\gamma}^{u_{{\cal
P}}+3u_{v}-1}\sim\dot{\gamma}^{u_{w}},$ (6)

with $u_{w}=0.565$, and note that this is in excellent agreement with
$q_{2}\approx 0.567$ from the fit of $\sigma(\phi,\dot{\gamma})$ to Eq. (3).

We now turn to the magnitude of the contribution due to the slow process. We
formally split ${\cal P}(v)$ into two parts for the slow and the fast
processes, ${\cal P}(v)={\cal P}_{s}(v)+{\cal P}_{f}(v)$. With this splitting
the dissipation from the slow process becomes ${\cal D}_{s}=k_{d}\int{\cal
P}_{s}(v)v^{2}dv$. We get $\sigma_{s}=(N/V\dot{\gamma}){\cal D}_{s}$, which
means that $\sigma_{s}$ is determined from the dissipation.

To demonstrate that $\sigma_{s}$ from the dissipation is indeed equal to the
correction-to-scaling term $a_{2}\dot{\gamma}^{q_{2}}$, in Eq. (3), we use the
reasoning behind Eq. (5) with $f_{s}(x)\equiv{\cal P}_{s}(v)/{\cal P}_{p}$, to
get

$\sigma_{s}=\frac{N}{V}k_{d}W_{p}I_{2},$ (7)

where $I_{2}\equiv\int_{0}^{\infty}f_{s}(x)x^{2}dx$. We find that the choice
$I_{2}=3.4$ gives a good agreement at $\phi_{J}$ between $\sigma_{s}$ and the
second term of Eq. (3). In Ref. joi (2022) it is shown that it is possible to
construct a reasonable $f_{s}(x)$ that is equal to $f(x)$ below the peak,
decays exponentially at larger $v$, and gives $I_{2}=3.4$. This shows that not
only the $\dot{\gamma}$-dependence, but also the magnitude, of the correction-
to-scaling term is consistent with it being due to the rescaled $f_{s}(x)$.

In Sec. III E of Ref. joi (2022) we show this kind of analysis for data from
the hard disk limit at $\phi<\phi_{J}$ and show that it works well and gives
consistent results with the results here, through analyses from different
$\dot{\gamma}$ at $\phi_{J}$.

We now turn to analyzing the velocity correlations and then first note that
the very wide distribution of particle velocities is due to the fact that the
particle velocity is $v_{i}=f_{i}/k_{d}$, where the net force
$f_{i}\equiv\sum_{j}f_{ij}$ is usually much smaller than the typical contact
force, $f_{ij}$, since the contact forces usually almost balance each other
out. In cases where the contact forces fail to balance each other out, as e.g.
in Fig. 3(a) where the dark gray particle is squeezed between the two
contacting light gray particles, this may give an unusually large net force
and thereby a high velocity. The fastest particles are the ones with only two
contacting particles but in the Supplemental Material we show that they
nevertheless give only about 7% of the total dissipation and that most of the
dissipation is due to (short) chains of fast particles. We also argue that the
same mechanism that gives the fast particle in Fig. 3(a) also gives these
short chains of fast particles.

In Ref. Charbonneau  _et al._ (2015) it was found that the theoretically
expected force distribution was obtained if contacts that were related to
localized configurations were not include in the calculation, and it was
further argued that these contacts were due to buckler configurations. We note
the similarity of Fig. 3(a) and the buckler configuration shown in Ref.
Charbonneau  _et al._ (2015) and remark that particles that are irrelevant in
the approach of Ref. Charbonneau  _et al._ (2015) are here found to be
significant. Fig. 3(a) also suggests a possible relation between some of the
fast particles and irreversible contact changes in quasistatic shearing Morse
_et al._ (2020); Tuckman  _et al._ (2020).

It is well known Pouliquen (2004); Lechenault  _et al._ (2008); Heussinger
_et al._ (2010) that the dynamics becomes increasingly collective as jamming
is approached and a recent paper has revealed a rich behavior of the velocity
correlations Olsson and Teitel (2020). A large correlation length is what one
would expect when particles behave as a slowly moving fluid, but it is less
clear what to expect for high velocity particles, as in Fig. 3(a), which move
erratically because of squeezing. To answer this question Fig. 3(b) shows the
velocity correlation
$g(x)=[\left<v_{\nearrow}(0)v_{\searrow}(x\hat{x})\right>+\left<v_{\searrow}(0)v_{\nearrow}(x\hat{x})\right>]/(\mathbf{v}^{2}/2)$
which is a measure of the rotation of the non-affine velocity field Olsson and
Teitel (2020). We show both $g(x)$ from all particles and the contributions to
$g(x)$ from different sets of particles. We identify the velocities as “low”
or “high” according to the threshold velocity $v_{50}$, chosen such that 50%
of the dissipation is due to particles with $v<v_{50}$. This is similar to the
splitting into slow and fast processes, but also different since ${\cal
P}_{s}(v)$ and ${\cal P}_{f}(v)$ are overlaping, with no sharp threshold
velocity. Fig. 3(b) shows that the full $g(x)$ at $\phi=0.8434$ and
$\dot{\gamma}=10^{-7}$ decays exponentially with a length $\xi\approx 19.4$
and that it is the low velocity particles that strongly dominate the large
distance correlation; $g_{hh}(x)$ from two high velocity particles,
contributes less than 1% to the total $g(x)$, for large $x$. We belive that
this non-zero value of $g_{hh}(x)$ is because the fast particles also get a
contribution to their velocities from the slow process since they are embedded
in a set of particles that behaves as a slowly moving fluid.

Figure 3: Behaviors of fast particles. Panel (a) which is a snapshot at
$\phi=0.800$ shows a particle with velocity $v/\left<v\right>\approx 8.5$
which has a high velocity because it is not in a force-balanced state but is
squeezed between two other particles, shown by light gray. Panel (b) shows the
splitting of the velocity correlation function according to the low or high
velocity of the colliding particles: $g_{\mathrm{ll}}$ for two low velocity
particles, $g_{\mathrm{hh}}$ for two high velocity particles, and
$g_{\mathrm{lh}}$ due to one particle with low and one with high velocity. The
key message is that the contribution to the total $g(x)$ from two high
velocity particles is very small.

This therefore suggests that the large correlation length is due to the slow
process, only, and this is a finding with profound consequences as it suggests
that one process is responsible for the correlations whereas another process
is behind the leading divergence in the viscosity, which is at odds with usual
critical phenomena. The link between viscosity and correlation length would
therefore seem to be an indirect one, only, and it appears that there exists
some, as yet unknown, mechanism that connects the slow and the fast processes
together.

Figure 4: Finite size scaling at $\phi_{J}\approx 0.8434$. Panel (a) is the
raw data $\sigma(\phi_{J},\dot{\gamma},L)$ for $\dot{\gamma}=10^{-8}$ through
$2\times 10^{-5}$, with symbols as in Fig. 1(a), for $N=32$ through 262144.
Panel (b) shows that $[\sigma-a_{1}\dot{\gamma}^{q}]/\dot{\gamma}^{q_{2}}$
collapses when plotted vs $L\dot{\gamma}^{1/z}$, in agreement with Eq. (9). We
here use $1/z=0.26$, assuming $1/z\nu=0.26$ Olsson and Teitel (2011) and
$\nu=1$ Olsson and Teitel (2020).

The lack of large distance correlations of the fast particles has consequences
also for the finite size dependence, examined in simulations with different
$N\propto L^{2}$. To include the $L$-dependence in the critical scaling
analysis one adds $b/L$ as an additional argument to the scaling functions of
Eq. (1), and Eq. (3) then becomes

$\sigma(\phi_{J},\dot{\gamma},L)=\dot{\gamma}^{q}\tilde{g}_{\sigma}(L\dot{\gamma}^{1/z})+\dot{\gamma}^{q_{2}}\tilde{h}_{\sigma}(L\dot{\gamma}^{1/z}).$
(8)

This means that one would expect $\sigma(\phi_{J},\dot{\gamma},L)$ to be the
sum of two $L$-dependent functions, respectively approaching $a_{1}$ and
$a_{2}$ for large $L$, with different prefactors, $\dot{\gamma}^{q}$ and
$\dot{\gamma}^{q_{2}}$. It does however turn out that the data, shown in Fig.
4(a), fit very well to a simpler expression without any finite size dependence
in the first term,

$\sigma(\phi_{J},\dot{\gamma},L)=a_{1}\dot{\gamma}^{q}+\dot{\gamma}^{q_{2}}\tilde{h}_{\sigma}(L\dot{\gamma}^{1/z}).$
(9)

This is demonstrated in Fig. 4(b) which shows that
$[\sigma(\phi_{J},\dot{\gamma},L)-a_{1}\dot{\gamma}^{q}]/\dot{\gamma}^{q_{2}}$,
with $q$, $q_{2}$, and $a_{1}$ the same as in Eq. (3), collapses onto a single
function when plotted vs $L\dot{\gamma}^{1/z}$. Even though there is nothing
in the formalism that excludes the possibility that the function
$\tilde{g}_{\sigma}(L\dot{\gamma}^{1/z})$ could be a constant $=a_{1}$, the
absence of a clear finite size dependence for the leading diverging term is
clearly at odds with the ordinary behavior in critical phenomena. It is
however entirely in accordance with the absence of correlations across large
distances in Fig. 3(b) for the particles with higher velocities, which are the
ones that dominate the $\dot{\gamma}^{q}$-term.

To summarize, we provide strong evidence that shear-driven jamming is governed
by two processes with different properties: The fast process is responsible
for the leading divergence of the shear viscosity whereas the slow process is
behind the diverging correlation length. The absence of a direct coupling
between these diverging quantities suggests that shear-driven jamming is an
unusual kind of critical phenomenon.

###### Acknowledgements.

I thank S. Teitel for many discussions and suggestions on the manuscript. The
computations were enabled by resources provided by the Swedish National
Infrastructure for Computing (SNIC) at High Performance Computer Center North,
partially funded by the Swedish Research Council through grant agreement no.
2018-05973.

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# TripoSR: Fast 3D Object Reconstruction from a Single Image

Dmitry Tochilkin1 David Pankratz1 Zexiang Liu2 Zixuan Huang1 Adam Letts1
Yangguang Li2 Ding Liang2 Christian Laforte1 Varun Jampani1∗ Yan-Pei Cao2∗

1Stability AI, 2Tripo AI

###### Abstract

This technical report introduces TripoSR, a 3D reconstruction model leveraging
transformer architecture for fast feed-forward 3D generation, producing 3D
mesh from a single image in under 0.5 seconds. Building upon the LRM [11]
network architecture, TripoSR integrates substantial improvements in data
processing, model design, and training techniques. Evaluations on public
datasets show that TripoSR exhibits superior performance, both quantitatively
and qualitatively, compared to other open-source alternatives. Released under
the MIT license, TripoSR is intended to empower researchers, developers, and
creatives with the latest advancements in 3D generative AI.

Model: https://huggingface.co/stabilityai/TripoSR
Code: https://github.com/VAST-AI-Research/TripoSR
Demo: https://huggingface.co/spaces/stabilityai/TripoSR

Figure 1: We present TripoSR, a 3D reconstruction model that reconstructs
high-quality 3D from single images in under 0.5 seconds. Our model achieves
state-of-the-art performance and generalizes to objects of various types and
input images across different domains.

**footnotetext: Equal advising.

##  1 Introduction

The landscape of 3D Generative AI has witnessed a confluence of developments
in recent years, blurring the lines between 3D reconstruction from single or
few views and 3D generation [33, 34, 29, 3, 11, 35, 13, 17, 9]. This
convergence has been significantly accelerated by the introduction of large-
scale public 3D datasets [4, 5] and advances in generative model
architectures. Comprehensive reviews of these technologies can be found in the
literature such as [15] and [22].

To overcome the scarcity of 3D training data, recent efforts have explored
utilizing 2D diffusion models to create 3D assets from text prompts [20, 27,
21] or input images [17, 23]. DreamFusion [20], a notable example, introduced
score distillation sampling (SDS), employing a 2D diffusion model to guide the
optimization of 3D models. This approach represents a pivotal strategy in
leveraging 2D priors for 3D generation, achieving breakthroughs in generating
detailed 3D objects. However, these methods typically face limitations with
slow generation speed, due to the extensive optimization and computational
demands, and the challenge of precisely controlling the output models.

On the contrary, feed-forward 3D reconstruction models achieve significantly
higher computational efficiency [8, 7, 32, 25, 19, 12, 11, 14, 31, 26, 17, 35,
13, 28, 24]. Several recent approaches [11, 14, 31, 26, 17, 35, 13, 28, 24]
along this direction have shown promise in scalable training on diverse 3D
datasets. These approaches facilitate rapid 3D model generation through fast
feed-forward inference and are potentially more capable of providing precise
control over the generated outputs, marking a notable shift in the efficiency
and applicability of these models.

In this work, we introduce TripoSR model for fast feed-forward 3D generation
from a single image that takes less than 0.5 seconds on an A100 GPU. Building
upon the LRM [11] architecture, we introduce several improvements in terms of
data curation and rendering, model design and training techniques.
Experimental results demonstrate superior performance, both quantitatively and
qualitatively, compared to other open-source alternatives. Figure 1 shows some
sample results of the TripoSR. TripoSR is made available under the MIT
license, accompanied by source code, the pretrained model, and an interactive
online demo. The release aims to enable researchers, developers, and creatives
to advance their work with the latest advancements in 3D generative AI,
promoting progress within the wider domains of AI, computer vision, and
computer graphics. Next, we introduce the technical advances in our TripoSR
model, followed by the quantitative and qualitative results on two public
datasets.

##  2 TripoSR: Data and Model Improvements

The design of TripoSR is based on the LRM [11], with a series of technical
advancements in data curation, model and training strategy. We now give an
overview of the model followed by our technical improvements.

###  2.1 Model Overview

Similar to LRM [11], TripoSR leverages the transformer architecture and is
specifically designed for single-image 3D reconstruction. It takes a single
RGB image as input and outputs a 3D representation of the object in the image.
The core of TripoSR includes components: an image encoder, an image-to-
triplane decoder, and a triplane-based neural radiance field (NeRF).

Parameter | Value
---|---
Image Tokenizer | image resolution | $512\times 512$
patch size | $16$
# attention layers | $12$
# feature channels | $768$
Triplane Tokenizer | # tokens | $32\times 32\times 3$
# channels | $16$
Backbone | # channels | $1024$
attention layers | $16$
# attention heads | $16$
attention head dim | $64$
cross attention dim | $768$
Triplane Upsampler | factor | $2$
# input channels | $1024$
# output channels | $40$
output shape | $64\times 64\times 40$
NeRF MLP | width | $64$
# layers | $10$
activation | $\mathrm{SiLU}$
Renderer | # samples per ray | $128$
radius | $0.87$
density activation | $\mathrm{exp}$
density bias | $-1.0$
Training | learning rate | $4\mathrm{e}{-4}$
optimizer | AdamW
lr scheduler | CosineAnnealingLR
# warm-up steps | $2,000$
$\lambda_{\mathrm{LPIPS}}$ | $2.0$
$\lambda_{\mathrm{mask}}$ | $0.05$
Table 1: Model configuration of TripoSR.

The image encoder is initialized with a pre-trained vision transformer model,
DINOv1 [1], which projects an RGB image into a set of latent vectors. These
vectors encode the global and local features of the image and include the
necessary information to reconstruct the 3D object.

The subsequent image-to-triplane decoder transforms the latent vectors onto
the triplane-NeRF representation [2]. The triplane-NeRF representation is a
compact and expressive 3D representation, well-suited for representing objects
with complex shapes and textures. Our decoder consists of a stack of
transformer layers, each with a self-attention layer and a cross-attention
layer. The self-attention layer allows the decoder to attend to different
parts of the triplane representation and learn relationships between them. The
cross-attention layer allows the decoder to attend to the latent vectors from
the image encoder and incorporate global and local image features into the
triplane representation. Finally, the NeRF model consists of a stack of
multilayer perceptrons (MLPs), which are responsible for predicting the color
and density of a 3D point in space.

Instead of conditioning the image-to-triplane projection on camera parameters,
we have opted to allow the model to “guess” the camera parameters (both
extrinsics and intrinsics) during training and inference. This is to enhance
the model’s robustness to in-the-wild input images at inference time. By
foregoing explicit camera parameter conditioning, our approach aims to
cultivate a more adaptable and resilient model capable of handling a wide
range of real-world scenarios without the need for precise camera information.

The architecture’s main parameters, such as the number of layers in the
transformer, the dimensions of the triplanes, the specifics of the NeRF model,
and the main training configurations, are detailed in Table 1. Compared to LRM
[11], TripoSR introduces several technical improvements which we discuss next.

###  2.2 Data Improvements

Recognizing the critical importance of data, we have incorporated two
improvements in our training data collection:

  * •

Data Curation: By selecting a carefully curated subset of the Objaverse [4]
dataset, which is available under the CC-BY license, we have enhanced the
quality of training data.

  * •

Data Rendering: We have adopted a diverse array of data rendering techniques
that more closely emulate the distribution of real-world images, thereby
enhancing the model’s ability to generalize, even when trained exclusively
with the Objaverse dataset.

###  2.3 Model and Training Improvements

Our adjustments aim to boost both the model’s efficiency and its performance.

Triplane Channel Optimization. The configuration of channels within the
triplane-NeRF representation plays an important role in managing the GPU
memory footprint during both training and inference, due to the high
computational cost of volume rendering. Moreover, the channel count
significantly influences the model’s capacity for detailed and high-fidelity
reconstruction. In pursuit of an optimal balance between reconstruction
quality and computational efficiency, experimental evaluations led us to adopt
a configuration of $40$ channels. This choice enables the use of larger batch
sizes and higher resolutions during the training phase, while concurrently
minimizing the memory requirements during inference.

Mask Loss. We incorporated a mask loss function during training that
significantly reduces “floater” artifacts and improves the fidelity of
reconstructions:

$\displaystyle\mathcal{L}_{\mathrm{mask}}(\boldsymbol{\hat{M}}_{v},\boldsymbol{M}^{\mathit{GT}}_{v})$
$\displaystyle=\mathrm{BCE}(\boldsymbol{\hat{M}}_{v},\boldsymbol{M}^{\mathit{GT}}_{v}),$
(1)

where $\boldsymbol{\hat{M}}_{v}$ and $\boldsymbol{M}^{\mathit{GT}}_{v}$ are
rendered and ground-truth mask images of the $v$-th supervision view,
respectively. The full training loss we minimized during training is:

$\begin{split}\mathcal{L}_{\mathrm{recon}}(\boldsymbol{I})&=\frac{1}{V}{\sum_{v=1}^{V}}\left(\mathcal{L}_{\mathrm{MSE}}(\boldsymbol{\hat{I}}_{v},\boldsymbol{I}^{\mathit{GT}}_{v})\right.\\\
&+{\lambda_{\mathrm{LPIPS}}}\mathcal{L}_{\mathrm{LPIPS}}(\boldsymbol{\hat{I}}_{v},\boldsymbol{I}^{\mathit{GT}}_{v})\\\
&\left.+{\lambda_{\mathrm{mask}}}\mathcal{L}_{\mathrm{mask}}(\boldsymbol{\hat{M}}_{v},\boldsymbol{M}^{\mathit{GT}}_{v})\right)\end{split}$
(2)

Local Rendering Supervision. Our model fully relies on rendering losses for
supervision, thereby imposing a need for high-resolution rendering for our
model to learn detailed shape and texture reconstructions. However, rendering
and supervising at high resolutions (e.g., $512\times 512$ or higher) can
overwhelm computational and GPU memory loads. To circumvent this issue, we
render $128\times 128$-sized random patches from the original $512\times 512$
resolution images during training. Crucially, we increase the likelihood of
selecting crops that cover foreground regions, thereby placing greater
emphasis on the areas of interest. This importance sampling strategy ensures
faithful reconstructions of object surface details, effectively balancing
computational efficiency and reconstruction granularity.

Figure 2: We outperform SOTA methods for 3D reconstruction while achieving
fast inference time. In the figure, F-Score with threshold 0.1 is averaged
over GSO [6] and OmniObject3D [30]. Figure 3: Qualitative results. We compare
TripoSR output meshes to other SOTA methods on GSO and OmniObject3D (first
four columns are from GSO [6], last two are from OmniObject3D [30]). Our
reconstructed 3D shapes and textures achieve significantly higher quality and
better details than previous state-of-the-art methods.

##  3 Results

We quantitatively and qualitatively compare TripoSR to previous state-of-the-
art methods using two different datasets with 3D reconstruction metrics.

Evaluation Datasets. We curate two public datasets, GSO [6] and OmniObject3D
[30], for evaluations. We identify that both datasets include many simple-
shaped objects (e.g., box, sphere or cylinder) and can thus cause high
validation bias towards these simple shapes. Therefore we manually filter the
datasets and select around 300 objects from each dataset to make sure they
form a diverse and representative collection of common objects.

3D Shape Metrics. We extract the isosurface using Marching Cubes [18] to
convert implicit 3D representations (such as NeRF) into meshes. We sample 10K
points from these surfaces to calculate the Chamfer Distance (CD) and F-score
(FS). Considering that some methods are not capable of predicting view-centric
shapes, we use a brute-force search approach to align the predictions with the
ground truth shapes. We linearly search the rotation angle by optimizing for
the lowest CD and further employ the Iterative Closest Point (ICP) method to
refine the alignment.

Method |  CD$\downarrow$ |<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---|---|---
One-2-3-45 [16] | 0.227 | 0.382 | 0.630 | 0.878
ZeroShape [13] | 0.160 | 0.489 | 0.757 | 0.952
TGS [35] | 0.122 | 0.637 | 0.846 | 0.968
OpenLRM [10] | 0.180 | 0.430 | 0.698 | 0.938
TripoSR (ours) | 0.111 | 0.651 | 0.871 | 0.980

Table 2: Quantitative comparison of different techniques on GSO [6] validation
set, where CD and FS refer to Chamfer Distance and F-score respectively.

Method |  CD$\downarrow$ |<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---|---|---
One-2-3-45 [16] | 0.197 | 0.445 | 0.698 | 0.907
ZeroShape [13] | 0.144 | 0.507 | 0.786 | 0.968
TGS [35] | 0.142 | 0.602 | 0.818 | 0.949
OpenLRM [10] | 0.155 | 0.486 | 0.759 | 0.959
TripoSR (ours) | 0.102 | 0.677 | 0.890 | 0.986

Table 3: Quantitative comparison of different techniques on OmniObject3D [30]
validation set, where CD and FS refers to Chamfer Distance and F-score
respectively.

Quantitative Comparisons. We compare TripoSR with the existing state-of-the-
art baselines on 3D reconstruction that use feed-forward techniques, including
One-2-3-45 [16], TriplaneGaussian (TGS) [35], ZeroShape [13] and OpenLRM
[10]111We use the openlrm-large-obj-1.0 model.. As shown in Table 2 and Table
3, our TripoSR significantly outperforms all the baselines, both in terms of
CD and FS metrics, achieving the new state-of-the-art performance on this
task.

Performance vs. Runtime. Another key advantage of TripoSR is its inference
speed. It takes around 0.5 seconds to produce a 3D mesh from a single image on
an NVIDIA A100 GPU. Figure 2 shows a 2D plot of different techniques with
inference times along the x-axis and the averaged F-Score along the y-axis.
The plot shows that TripoSR is among the fastest networks, while also being
the best-performing feed-forward 3D reconstruction model.

Visual Results. We further show the qualitative results of different
approaches in Figure 3. Because some methods do not reconstruct textured
meshes, we render TripoSR reconstructions both with and without vertex color
for a better comparison. As shown in the figure, ZeroShape tends to predict
over-smoothed shapes. TGS reconstructs more surface details but these details
sometimes do not align with the input. Moreover, both ZeroShape and TGS cannot
output textured meshes directly 222TGS leverages 3DGS to represent 3D objects.
We follow the paper and utilize their auxiliary point cloud outputs to
reconstruct the surface. However, it is non-trivial to reconstruct textures on
meshes, (e.g., directly taking vertex colors from the nearest Gaussian leads
to noisy textures).. On the other hand, One-2-3-45 and OpenLRM predict
textured meshes, but their estimated shapes are often inaccurate. Compared to
these baselines, TripoSR demonstrates a high reconstruction quality for both
shape and texture. Our model not only captures a better overall 3D structure
of the object, but also excels at modeling several intricate details.

##  4 Conclusion

In this report, we present an open-source feedforward 3D reconstruction model,
TripoSR. The core of our model is a transformer-based architecture developed
upon the LRM network [11], together with substantial technical improvements
along multiple axes. Evaluated on two public benchmarks, our model
demonstrates state-of-the-art reconstruction performance with high
computational efficiency. We hope TripoSR empowers researchers and developers
in developing more advanced 3D generative AI models.

## Acknowledgements

Tripo AI. We extend our sincere gratitude to Yuan-Chen Guo and Zi-Xin Zou for
their critical roles in coding, demo development, and experimentation. We also
thank Peng Wang for his insightful discussions, Dehu Wang for preparing the
datasets for model validation, and Sienna Huang for managing communication
within our collaboration.

Stability AI. We thank Emad Mostaque, Anel Islamovic, Bryce Wilson, Ana
Guillen, Adam Chen, Christian Dowell and Ella Irwin for their help in various
aspects of the model development, collaboration and release. We also thank
Vikram Voleti for helpful discussions and Eric Courtemanche for his help with
visual results.

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