hoomd.hpmc.compute¶
Overview
Compute the free volume available to a test particle. 

Compute the scale distribution function via volume perturbations. 
Details
Compute properties of hard particle configurations.
The HPMC compute classes analyze the system configuration and provide results
as loggable quantities for use with hoomd.logging.Logger
or by direct access
via the Python API. FreeVolume
computes the free volume available to small
particles, such as depletants, and SDF
computes the pressure in system of
convex particles with a fixed box size.
 class hoomd.hpmc.compute.FreeVolume(test_particle_type, num_samples)¶
Compute the free volume available to a test particle.
 Parameters:
FreeVolume
computes the free volume in the simulation state available to a given test particle shape using Monte Carlo integration. Use it in combination withhoomd.hpmc.integrate.HPMCIntegrator
, which defines the particle shape parameters. Particles oftest_particle_type
may or may not be present in the simulation state.FreeVolume
generatesnum_samples
(\(n_\mathrm{samples}\)) trial particle configurations with positions \(\vec{r}^t_j\) uniformly distributed in the simulation box, and orientations \(\mathbf{q}^t_j\) uniformly distributed among rotations matching the box dimensionality.FreeVolume
counts the number of successful samples that do not overlap particles in the simulation state:\[n_\mathrm{success} = \sum_{j=1}^{n_\mathrm{samples}} \prod_{i=0}^{N_\mathrm{particles}1} \prod_{\vec{A} \in B_\mathrm{images}} \left[ \mathrm{overlap}\left( S_i(\mathbf{q}_i), S_t(\mathbf{q}^t_j, \vec{r}^t_j  (\vec{r}_i + \vec{A})) \right) = \emptyset \right]\]where \(\mathrm{overlap}\) is the shape overlap function defined in
hoomd.hpmc.integrate
, \(S_i\) is the shape of particle \(i\), \(S_t\) is the shape of the test particle, \(\vec{A} = h\vec{a}_1 + k\vec{a}_2 + l\vec{a}_3\) is a vector that translates by periodic box images, the set of box images includes all image vectors necessary to find overlaps between particles in the primary image with particles in periodic images, and the square brackets denote the Iverson bracket.The free volume \(V_\mathrm{free}\) is given by:
\[V_\mathrm{free} = \frac{n_\mathrm{success}} {n_\mathrm{samples}} V_\mathrm{box}\]where \(V_\mathrm{box}\) is the volume of the simulation box (or area in 2D).
Note
FreeVolume
respects the HPMC integrator’sinteraction_matrix
.Mixed precision
FreeVolume
uses reduced precision floating point arithmetic when checking for particle overlaps in the local particle reference frame.Box images
On CPU devices,
FreeVolume
does not apply the minimum image convention. It supports small boxes where particles may overlap with nonprimary images of other particles, including self overlap. On GPU devices,FreeVolume
applies the minimum image convention.Examples:
fv = hoomd.hpmc.compute.FreeVolume(test_particle_type='B', num_samples=1000)
 class hoomd.hpmc.compute.SDF(xmax, dx)¶
Compute the scale distribution function via volume perturbations.
 Parameters:
SDF
computes the probability distributions \(s_{\mathrm{comp}}(x)\) and \(s_{\mathrm{exp}}(x)\) of particles overlapping as a function of separation for compressive and expansive perturbations, respectively. It estimates \(s_{\mathrm{comp}}(x)\) and \(s_{\mathrm{exp}}(x)\) numerically by computing histograms with \(\lfloor x_\mathrm{max}/ \delta x \rfloor\) bins of widthdx
(\(\delta x\)).See also
Anderson 2016 describes the theory relating
SDF
to the system pressure and its implementation in HOOMDblue. Eppenga and Frenkel 1984 present a derivation relating the scale distribution function to the system pressure for hard, convex particles. Allen 2006 describes the theory for calculating the pressure in systems with discontinuous potential energy functions. The expansive perturbations are based on theory described in de Miguel and Jackson.Implementation
SDF
constructs two histograms, one for compressive volume perturbations and one for expansive volume perturbations. These histograms represent \(s_{\mathrm{comp}}(x)\) and \(s_{\mathrm{exp}}(x)\). The following discussion applies to compressive volume perturbations and the computation of \(s_{\mathrm{comp}}(x)\); the computation of \(s_{\mathrm{exp}}(x)\) proceeds similarly as noted throughout the description.For each pair of particles \(i\) and \(j\)
SDF
scales the particle separation vector by the factor \((1 \pm x)\) (\(+\) for expansive perturbations, \(\) for compressive perturbations) and finds the smallest positive value of \(x\) leading to either an overlap of the particle shapes (a “hard overlap”) or a discontinuous change in the pair energy \(U_{\mathrm{pair},ij}\) (a “soft overlap”). For compressive perturbations:\[\begin{split}x_{ij}(\vec{A}) = \min \{ & x \in \mathbb{R}_{> 0} : \\ & \mathrm{overlap}\left( S_i(\mathbf{q}_i), S_j(\mathbf{q}_j, (1x)(\vec{r}_j  (\vec{r}_i + \vec{A}))) \right) \ne \emptyset \\ &\lor \\ & U_{\mathrm{pair},ij}((1x)(\vec{r}_j  (\vec{r}_i + \vec{A})), \mathbf{q}_i, \mathbf{q}_j) \ne U_{\mathrm{pair},ij}(\vec{r}_j  (\vec{r}_i + \vec{A}), \mathbf{q}_i, \mathbf{q}_j) \\ \} &\end{split}\]where \(\mathrm{overlap}\) is the shape overlap function defined in
hoomd.hpmc.integrate
, \(S_i\) is the shape of particle \(i\), and \(\vec{A} = h\vec{a}_1 + k\vec{a}_2 + l\vec{a}_3\) is a vector that translates by periodic box images. For expansive perturbations,\[\begin{split}x_{ij}(\vec{A}) = \max \{ & x \in \mathbb{R}_{< 0} : \\ & \mathrm{overlap}\left( S_i(\mathbf{q}_i), S_j(\mathbf{q}_j, (1+x)(\vec{r}_j  (\vec{r}_i + \vec{A}))) \right) \ne \emptyset \\ &\lor \\ & U_{\mathrm{pair},ij}((1+x)(\vec{r}_j  (\vec{r}_i + \vec{A})), \mathbf{q}_i, \mathbf{q}_j) \ne U_{\mathrm{pair},ij}(\vec{r}_j  (\vec{r}_i + \vec{A}), \mathbf{q}_i, \mathbf{q}_j) \\ \} &\end{split}\]For particle \(i\),
SDF
finds the the minimum (maximum for expansive perturbations) value \(x_i\). For compressive perturbations:\[x_i = \min \{ x_{ij} : \vec{A} \in B_\mathrm{images}, j \in [0,N_\mathrm{particles}) \}\]where the set of box images includes all image vectors necessary to find overlaps between particles in the primary image with particles in periodic images. For expansive perturbations:
\[x_i = \max \{ x_{ij} : \vec{A} \in B_\mathrm{images}, j \in [0,N_\mathrm{particles}) \}\]SDF
adds a single count to each histogram for each particle \(i\), weighted by a factor that is a function of the change in energy upon overlap. For compressive perturbations:\[s_{\mathrm{comp}}(x + \delta x/2) = \frac{1}{N_\mathrm{particles} \cdot \delta x} \sum_{i=0}^{N_\mathrm{particles}1} [x \le x_i < x + \delta x] \cdot (1  \exp(\beta \Delta U_{i}))\]where \(\Delta U_{i}\) is the change in energy associated with the first overlap involving particle \(i\) (\(\infty\) for hard overlaps), the square brackets denote the Iverson bracket, and \(s_{\mathrm{comp}}(x + \delta x/2)\) is evaluated for \(\{ x \in \mathbb{R}, 0 \le x < x_\mathrm{max}, x = k \cdot \delta x, k \in \mathbb{Z}^* \}\) for compressive perturbations. For expansive perturbations,
\[s_{\mathrm{exp}}(x  \delta x/2) = \frac{1}{N_\mathrm{particles} \cdot \delta x} \sum_{i=0}^{N_\mathrm{particles}1} [x  \delta x \le x_i < x] \cdot (1  \exp(\beta \Delta U_{i}))\]where \(s_{\mathrm{exp}}(x  \delta x/2)\) is evaluated for \(\{ x \in \mathbb{R}, x_\mathrm{max} < x \le 0, x = (k  \lfloor x_\mathrm{max} / \delta x \rfloor + 1) \cdot \delta x, k \in \mathbb{Z}^* \}\).
Pressure
The pressure \(P\) is related to the onesided limits \(s_{\mathrm{comp}}(0+)\) and \(s_{\mathrm{exp}}(0)\), computed by fitting and extrapolating \(s_{\mathrm{comp}}\) and \(s_{\mathrm{exp}}\) to \(x = 0\).
\[\beta P = \rho \left( 1 + \frac{s_{\mathrm{comp}}(0+)  s_{\mathrm{exp}}(0)}{2d} \right)\]where \(d\) is the dimensionality of the system, \(\rho\) is the number density, and \(\beta = \frac{1}{kT}\). This measurement of the pressure is inherently noisy due to the nature of the sampling. Average
betaP
over many timesteps to obtain accurate results.Assuming particle diameters are ~1, these parameter values typically achieve good results:
xmax = 0.02
dx = 1e4
In systems near densest packings,
dx=1e5
may be needed along with smallerxmax
. Check that \(\sum_k s_\mathrm{comp}(x_k) \cdot dx \approx 0.5\).Important
SDF
samples pair configurations at discrete separations. Therefore, the computed pressure is correct only for potentials with constant values and step discontinuities.Note
SDF
always runs on the CPU.Mixed precision
SDF
uses reduced precision floating point arithmetic when checking for particle overlaps in the local particle reference frame.Box images
SDF
does not apply the minimum image convention. It supports small boxes where particles may overlap with nonprimary images of other particles, including self overlap. xmax¶
Maximum x value at the right hand side of the rightmost bin \([\mathrm{length}]\).
 Type:
 property betaP¶
Beta times pressure in NVT simulations \(\left[ \mathrm{length}^{d} \right]\).
Uses a polynomial curve fit of degree 5 to estimate \(s_\mathrm{comp}(0+)\) (and \(s_\mathrm{exp}(0)\) if required) and computes the pressure via:
\[\beta P = \rho \left(1 + \frac{s_\mathrm{comp}(0+)}{2d} + \frac{s_\mathrm{exp}(0)}{2d} \right)\]where \(d\) is the dimensionality of the system, \(\rho\) is the number density, and \(\beta = \frac{1}{kT}\).
Attention
In MPI parallel execution,
betaP
is available on rank 0 only.betaP
isNone
on ranks >= 1.(
Loggable
: category=”scalar”) Type:
 property sdf_compression¶
\(s_\mathrm{comp}[k]\)  The scale distribution function for compression moves \([\mathrm{probability\ density}]\).
See also
x_compression
defines the bin center locations.Attention
In MPI parallel execution, the array is available on rank 0 only.
sdf_compression
isNone
on ranks >= 1.(
Loggable
: category=”sequence”) Type:
(N_bins,)
numpy.ndarray
offloat
)
 property sdf_expansion¶
\(s_\mathrm{exp}[k]\)  The scale distribution function for the expansion moves \([\mathrm{probability\ density}]\).
See also
x_expansion
defines the bin center locations..Attention
In MPI parallel execution, the array is available on rank 0 only.
sdf_expansion
isNone
on ranks >= 1.(
Loggable
: category=”sequence”) Type:
(N_bins,)
numpy.ndarray
offloat
)
 property x_compression¶
The x values at the center of each bin corresponding to the scale distribution function for the compressive perturbations \([\mathrm{length}]\).
(
Loggable
: category=”sequence”) Type:
(N_bins,)
numpy.ndarray
offloat
)
 property x_expansion¶
The x values at the center of each bin corresponding to the scale distribution function for the expansion moves \([\mathrm{length}]\).
(
Loggable
: category=”sequence”) Type:
(N_bins,)
numpy.ndarray
offloat
)