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   "source": [
    "# States\n",
    "\n",
    "A Riemann Problem is specified by the state of the material to the left and right of the interface. In this hydrodynamic problem, the state is fully determined by an [equation of state](eos_defns.html) and the variables\n",
    "\n",
    "$$\n",
    "  {\\bf U} = \\begin{pmatrix} \\rho_0 \\\\ v_x \\\\ v_t \\\\ \\epsilon \\end{pmatrix},\n",
    "$$\n",
    "\n",
    "where $\\rho_0$ is the rest-mass density, $v_x$ the velocity normal to the interface, $v_t$ the velocity tangential to the interface, and $\\epsilon$ the specific internal energy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Defining a state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In `r3d2` we define a state from an equation of state and the values of the key variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from r3d2 import eos_defns, State"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "eos = eos_defns.eos_gamma_law(5.0/3.0)\n",
    "U = State(1.0, 0.1, 0.0, 2.0, eos)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inside the notebook, the state will automatically display the values of the key variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\\begin{pmatrix} \\rho \\\\ v_x \\\\ v_t \\\\ \\epsilon \\end{pmatrix}= \\begin{pmatrix} 1.0000 \\\\ 0.1000 \\\\ 0.0000 \\\\ 2.0000 \\end{pmatrix}\\end{equation}"
      ],
      "text/plain": [
       "<r3d2.state.State at 0x7f031911c438>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adding a label to the state for output purposes requires an extra keyword:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\\begin{pmatrix} \\rho \\\\ v_x \\\\ v_t \\\\ \\epsilon \\end{pmatrix}_{L} = \\begin{pmatrix} 10.0000 \\\\ -0.3000 \\\\ 0.1000 \\\\ 5.0000 \\end{pmatrix}\\end{equation}"
      ],
      "text/plain": [
       "<r3d2.state.State at 0x7f031911c6a0>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U2 = State(10.0, -0.3, 0.1, 5.0, eos, label=\"L\")\n",
    "U2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reactive states\n",
    "\n",
    "If the state has energy available for reactions, that information is built into the equation of state. The definition of the equation of state changes: the definition of the state itself does not:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\\begin{pmatrix} \\rho \\\\ v_x \\\\ v_t \\\\ \\epsilon \\\\ q \\end{pmatrix}_{Reactive} = \\begin{pmatrix} 5.0000 \\\\ 0.1000 \\\\ 0.1000 \\\\ 2.0000 \\\\ 0.1000 \\end{pmatrix}\\end{equation}"
      ],
      "text/plain": [
       "<r3d2.state.State at 0x7f031911cb38>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q_available = 0.1\n",
    "t_ignition = 10.0\n",
    "Cv = 1.0\n",
    "eos_reactive = eos_defns.eos_gamma_law_react(5.0/3.0, q_available, Cv, t_ignition, eos)\n",
    "U_reactive = State(5.0, 0.1, 0.1, 2.0, eos_reactive, label=\"Reactive\")\n",
    "U_reactive"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional functions\n",
    "\n",
    "A state knows its own wavespeeds. Given a wavenumber (the left acoustic wave is `0`, the middle contact or advective wave is `1`, and the right acoustic wave is `2`), we have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Left wavespeed of first state is -0.6636390537799616\n",
      "Middle wavespeed of second state is -0.3\n",
      "Right wavespeed of reactive state is 0.7615771981098584\n"
     ]
    }
   ],
   "source": [
    "print(\"Left wavespeed of first state is {}\".format(U.wavespeed(0)))\n",
    "print(\"Middle wavespeed of second state is {}\".format(U2.wavespeed(1)))\n",
    "print(\"Right wavespeed of reactive state is {}\".format(U.wavespeed(2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A state will return the key *primitive* variables ($\\rho, v_x, v_t, \\epsilon$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Primitive variables of first state are [1.  0.1 0.  2. ]\n"
     ]
    }
   ],
   "source": [
    "print(\"Primitive variables of first state are {}\".format(U.prim()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A state will return all the variables it computes, which is $\\rho, v_x, v_t, \\epsilon, p, W, h, c_s$: the primitive variables as above, the pressure $p$, Lorentz factor $W$, specific enthalpy $h$, and speed of sound $c_s$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "All variables of second state are [1.         0.1        0.         2.         1.33333333 1.00503782\n",
      " 4.33333333 0.71611487]\n"
     ]
    }
   ],
   "source": [
    "print(\"All variables of second state are {}\".format(U.state()))"
   ]
  }
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